Algebra Assignments
This book allows a student or user to practice the following concepts: graphing transformations and finding real zeros of polynomials.
Table of Contents
- 1. Linear Function Transformations
- Linear Function Transformations
- 2. Quadratic Function Transformations
- Quadratic Function Transformations
- 3. Cubic Function Transformations
- Cubic Function Transformations
- 4. Absolute Function Transformations
- Absolute Value Function Transformations
- 5. Square Root Function Transformations
- Square Root Function Transformations
- 6. Rational or Reciprocal Function Transformations
- Rational or Reciprocal Function Transformations
- 7. Exponential Function Transformations
- Exponential Function Transformations
- 8. Logarithmic Function Transformations
- Logarithmic Function Transformations
- 9. Cube Root Function Transformations
- Cube Root Function Transformations
- 10. Finding Real Zeros of Polynomials
- Finding Real Zeros of Polynomials
Linear Function Transformations
Linear Function Transformations
Linear Function Transformation Exercise
[br][b][size=150]The linear function [color=#ff0000]y = x[/color], denoted by function g. [br][/size] [br][/b]The slope-intercept form is [color=#ff0000][b]y = mx + b[/b][/color], where m=slope and b=y-intercept of the function.[br][br] [size=150][color=#ff0000][b]Note[/b][/color][/size]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]m and b[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][color=#ff00ff][b] Note: You can zoom in or out with the mouse.[/b][br][/color]
Exercise 1
[size=150][b]Perform the following linear function transformation:[br][/b][/size][br][b]Vertical shift of 3 units up (y-intercept = 3). [br][br][/b] [color=#0000ff]The new function is [/color][b][color=#ff0000]y=x +3[/color][/b][color=#0000ff] , denoted by function f.[br] Set the slope of the function to m=1 by entering 1 for m.[br] Set the y-intercept of the function to b=3 by entering 3 for b. [br][/color][color=#ff00ff][b] Observe the transformation of the linear function.[/b][/color]
Exercise 2
[b]Perform the following linear function transformation:[br][/b][br][b]Vertical shift of 3 units down (y-intercept = -3). [br][br][/b] [color=#0000ff]The new function is [/color][b][color=#ff0000]y=x - 3[/color][/b] , [color=#0000ff]denoted by function f.[br] Set the slope of the function to m=1 by entering 1 for m.[br] Set the y-intercept of the function to b=-3 by entering -3 for b. [br][/color][color=#ff00ff][b] Observe the transformation of the linear function.[/b][/color]
Exercise 3
[b]Perform the following linear function transformation:[br][/b][br][b]Change slope of the linear function to 2. [br][br][/b] [color=#0000ff]The new function is [/color][b][color=#ff0000]y=2x[/color][/b] , [color=#0000ff]denoted by function f.[br] Set the slope of the function to 2 by entering 2 for m.[br] Set the y-intercept of the function to b=0 by entering 0 for b. [br][/color][color=#ff00ff][b] Observe the transformation of the linear function.[/b][/color]
Exercise 4
[b]Perform the following linear function transformation:[br][/b][b][br]Change slope of the linear function to - 2. [br][br][/b] [color=#0000ff]The new function is [/color][b][color=#ff0000]y= - 2x[/color][/b] ,[color=#0000ff]denoted by function f. [/color][br][color=#0000ff] Set the slope of the function to -2 by entering -2 for m.[br] Set the y-intercept of the function to b=0 by entering 0 for b. [br][/color][b][color=#ff00ff] Observe the transformation of the linear function.[/color][/b]
Exercise 5
[b][b]Perform the following linear function transformation:[br][/b][br]Graph a constant linear function by changing the slope of the [br] linear function to 0 with a y-intercept of 3.[br][br][/b] [color=#0000ff]The new function is [/color][b][color=#ff0000]y= 0 +3 = 3 [/color][/b] , [color=#0000ff]denoted by function f[/color]. [br][color=#0000ff] Set the slope of the function to 0 by entering zero for m.[br] Set the y-intercept of the function to 3 by entering 3 for b. [br][/color][color=#ff00ff][b] Observe the transformation of the linear function. [/b][/color][br]
Exercise 6
[b]Repeat this exercise as many times as desired until concept is mastered. [/b] [br][br] Use different values of [color=#ff0000][b]m and b[/b][/color].[br]
Quadratic Function Transformations
Quadratic Function Transformations
Quadratic Function Transformation Exercise
[b][size=150]The quadratic function is [color=#ff0000]y = x[sup]2[/sup][/color] , denoted by function g. [br][br][/size][/b]The transformed basic function is [b][color=#ff0000]y = a(bx - h)[sup]2[/sup] +k[/color][/b][br][b][color=#ff0000][size=150][br]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up. [br] [br][/b] The new function is [b][color=#ff0000]y=x[sup]2[/sup] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 2
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br] [br][/b] The new function is [b][color=#ff0000]y=x[sup]2[/sup] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 3
[b][size=150]Perform the following quadratic function transformation:[br][br][/size][/b][b]Horizontal shift of 3 units to the right. [br][br][/b] The new function is [b][color=#ff0000]y=(x-3)[sup]2[/sup][/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[br][/color][/b]
Exercise 4
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=(x+3)[sup]2[/sup][/color] [/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 5
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = (x-3)[sup]2[/sup] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 6
[b][size=150]Perform the following quadratic function transformation:[br][br][/size][/b][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = (x+3)[sup]2[/sup] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 7
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = (x - 3)[sup]2[/sup] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 8
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = (x + 3)[sup]2[/sup] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 9
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3.[br][br][/b] New function: [color=#ff0000] [b]y = 3 x[sup]2[/sup][/b] [/color] , denoted by function f.[br][br] [color=#0000ff]Set a=3. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 10
[b][size=150]Perform the following quadratic function transformation:[br][br][/size][/b][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3 x[sup]2[/sup][/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1/3. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][br][/color][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 11
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left [br] and a vertical stretch by a factor of 2 . [br][br][/b] New function: [b][color=#ff0000]y = 2(x + 3)[sup]2[/sup] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 12
[b][size=150]Perform the following quadratic function transformation:[br][br][/size][/b][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left, [br] a vertical shrink by a factor of 1/2 . [br][br][/b] New function: [b][color=#ff0000]y = 1/2(x + 3)[sup]2[/sup] + 3 [/color][/b], denoted by function f.[br] [color=#0000ff]Set a=2. Set b=1.[br][br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 13
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = (1/3x)[sup]2[/sup][/b] [/color] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][color=#ff00ff][b] Observe the transformation of the quadratic function.[/b][/color]
Exercise 14
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3.[br][br][/b] New function: [b][color=#ff0000]y = (3x)[sup]2[/sup][/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b = 3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 15
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - x[sup]2[/sup] [/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=-1. Set b = 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 16
[b][size=150]Perform the following quadratic function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = (-x)[sup]2[/sup][/color][/b] , denoted by function f.[br][br] [color=#0000ff] Set a=1. Set b= -1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the quadratic function.[/color][/b]
Exercise 17
[br][b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].[br][br][br]
Cubic Function Transformations
Cubic Function Transformations
Cubic Function Transformation Exercise
[b][size=150]The cubic function is [color=#ff0000]y = x[/color][color=#ff0000][sup]3 [/sup][/color] , denoted by function g. [br][br][/size][/b]The transformed basic function is [b][color=#ff0000]y = a(bx - h)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +k[/color][/b][br][br][b][color=#ff0000][size=150]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][color=#ff00ff][b] Observe the transformation of the cubic function.[/b][/color]
Exercise 2
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff].[/color][/b]
Exercise 3
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Horizontal shift of 3 units to the right. [br][br][/b] The new function is [b][color=#ff0000]y=(x-3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff]. [/color][/b]
Exercise 4
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=(x+3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]
Exercise 5
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b]y = [color=#ff0000](x-3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 6
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br] [br][/b] New function: [b][color=#ff0000]y = (x+3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 7
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = (x - 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = (x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b],denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3. [br][br][/b] New function: [color=#ff0000] [b]y = 3 [/b][/color][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=3. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]
Exercise 10
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3 [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1/3. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 11
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = (1/3x)[/b][/color][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 12
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3[br][/b] New function: [b][color=#ff0000]y = (3x)[/color][/b] [b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 13
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical stretch by a factor of 2. [br][br][/b] New function: [b][color=#ff0000]y = 2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 14
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical shrink by a factor of 1/2. [br][br][/b] New function: [b][color=#ff0000]y = 1/2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 15
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=-1. Set b = 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 16
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = (-x)[/color][/b][b][size=150][color=#ff0000][sup]3 [/sup][/color][/size][/b], denoted by function f.[br][br] [color=#0000ff] Set a=1. Set b = -1. [br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 17
[b]Repeat this exercise as many times as desired until concept is mastered. [/b] [br][br]Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].
Absolute Value Function Transformations
Absolute Value Function Transformations
Absolute Value Function Transformation Exercise
[b][size=150]The absolute value function is [color=#ff0000]y = |x|[/color] , denoted by function g. [br][br][/size][/b]The transformed basic function is [color=#ff0000][b]y = a|bx - h| +k[/b][/color].[br][br][b][color=#ff0000][size=150]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]|x|[/color][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the absolute value function.[/color][/b]
Exercise 2
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down.[br] [br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]|x|[/color][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 3
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]|x-3|[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 4
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]|x+3|[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 4
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]|x+3|[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 5
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b]y = [/b][b][color=#ff0000]|x-3|[/color][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 6
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y =|x+3| - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 7
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = |x - 3| - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3.[br][br][/b] New function: [color=#ff0000] [b]y = 3|x|[/b] [/color], denoted by function f.[br][br] [color=#0000ff]Set a=3. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = |x + 3| + 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents[/color] [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][color=#ff00ff][b] Observe the transformation of the absolute value function.[/b][/color]
Exercise 10
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3| x|[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1/3.[br] Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]absolute value [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 11
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = |1/3x|[/b] [/color], denoted by function f. [br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 12
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3.[br][br][/b] New function: [b][color=#ff0000]y = |3x|[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 13
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units, a horizontal shift of 3 units to the left[br] and a vertical stretch by a factor of 2. [br][br][/b] New function: [b][color=#ff0000]y = 2|x + 3| + 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[/color][color=#0000ff][br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 14
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Vertical shift of 3 units, a horizontal shift of 3 units to the left[br] and a vertical shrink by a factor of 1/2. [br][/b] [br] New function: [b][color=#ff0000]y = 1/2|x + 3| + 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 15
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - |x|[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a=-1. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 16
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = |-x|[/color][/b] , denoted by function f. [br][br] [color=#0000ff] Set a=1. Set b=-1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]absolute value [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 17
[b][size=150]Perform the following absolute value function transformation:[br][/size][/b][br][b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b].[/color]
Exercise 17
[br][b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b].[/color]
Square Root Function Transformations
Square Root Function Transformations
Square Root Function Transformation Exercise
[b][size=150]The square root function is[color=#ff0000] y= SQRT X, [/color]denoted by function g. [br][br][/size][/b]The transformed basic function is [b][color=#ff0000]y = a SQRT(bx - h)[sup]2[/sup] +k[/color][/b][br][br][b][color=#ff0000][size=150]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][color=#ff00ff][b]Note: You can zoom in or out with the mouse.[/b][/color]
Exercise 1
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=SQRT (x) +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the square root function.[/color][/b]
Exercise 2
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]SQRT (x)[/color][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 3
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. [br] [br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]SQRT (x -3)[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 4
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x+3)[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][b][color=#ff00ff]square root [/color][/b][color=#ff00ff]function.[/color][/b]
Exercise 5
[b][size=150]Perform the following square root function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b]y = [/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x-3) +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 6
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x+3) - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 7
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x - 3) - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x + 3) + 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1.[/color] [color=#0000ff]Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3.[br][br][/b] New function: [color=#ff0000] [b]y = 3 [/b][/color][b][color=#ff0000]SQRT ([/color][/b][color=#ff0000][b] x)[/b][/color], denoted by function f. [br][br] [color=#0000ff]Set a=3. Set b=1. [br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 10
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3 [/color][/b][b][color=#ff0000]SQRT ([/color][/b][b][color=#ff0000]x)[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a=1/3. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]square root [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 11
[b][size=150]Perform the following square root function transformation:[br][/size][/b][b][br]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = [/b][/color][b][color=#ff0000]SQRT [/color][/b][color=#ff0000][b](1/3x)[/b] [/color] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 12
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3[br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](3x)[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a =1. Set b=3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 13
[b][size=150]Perform the following square root function transformation:[br][br][/size][/b][b]Vertical shift of 3 units up. horizontal shift of 3 units to the left[br] and a vertical stretch by a factor of 2. [br][br][/b] New function: [b][color=#ff0000]y = 2[/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x + 3) + 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 14
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. horizontal shift of 3 units to the left[br] and a vertical shrink by a factor of 1/2. [br][br][/b] New function: [b][color=#ff0000]y = 1/2[/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](x + 3) + 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 15
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - [/color][/b][b][color=#ff0000]SQRT (x)[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=-1. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 16
[b][size=150]Perform the following square root function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = [/color][/b][b][color=#ff0000]SQRT [/color][/b][b][color=#ff0000](-x)[/color][/b] , denoted by function f.[br][br] [color=#0000ff] Set a=1. Set b=-1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]square root [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 17
[b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].
Rational or Reciprocal Function Transformations
Rational or Reciprocal Function Transformations
Rational or Reciprocal Function Transformation Exercise
[size=150][b]The rational or reciprocal function is [color=#ff0000]y[/color][color=#ff0000] = 1/[/color][/b][/size][size=150][color=#ff0000][b]x[/b][/color][b] , denoted by function g.[br] [br][/b][/size]The transformed basic function is [color=#ff0000][b]y = 1/(x - h) + k[br][/b][/color][b][color=#ff0000][size=150][br]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b]1/x [/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 2
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b]1/x [/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 3
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. [br][br][/b] The new function i [b][color=#ff0000]y=[/color][/b][b]1/(x - 3)[/b] , denoted by function f.[br][br][color=#0000ff] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 4
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=1/(x+3)[/color] [/b] , denoted by function f.[br][br][color=#0000ff] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 5
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = 1/(x-3) +3[/color][/b] , denoted by function f.[br][br][color=#0000ff] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 6
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = 1/(x+3) - 3 [/color][/b], denoted by function f.[br][br][color=#0000ff] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the rational function.[/color][/b]
Exercise 7
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = 1/(x - 3) - 3[/color][/b], denoted by function f.[br][br][color=#0000ff] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]rational [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = 1/(x+3) + 3[/color][/b], denoted by function f.[br][br][color=#0000ff] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]rational [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - 1/x [/color][/b] , denoted by function f.[br][br][color=#0000ff] Place a negative in front of the entire equation.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]rational [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 10
[b][size=150]Perform the following rational function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = 1/(-x )[/color][/b] , denoted by function f.[br][br] [color=#0000ff] Place a negative in front of the variable x.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]rational [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 11
[b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b] h and k[/b][/color].
Exponential Function Transformations
Exponential Function Transformations
Exponential Function Transformation Exercise
[b][size=150]The exponential function is [color=#ff0000]y = a[/color][/size][/b][b][size=150][color=#ff0000][sup]x[/sup][/color] , denoted by function g. [br] [/size][/b][b][size=150][color=#ff0000][math]\cdot[/math][/color][/size][/b][br]The transformed basic function is [b][size=150][color=#ff0000]y = b a[/color][/size][/b][b][size=150][color=#ff0000][sup]x+h[/sup][/color] [color=#ff0000]+ k with [/color][/size][size=150][color=#ff0000]a > 0, a [/color][color=#ff0000]≠ [/color][color=#ff0000]1[/color][/size][size=150][color=#ff0000].[/color][/size][/b][br][b][color=#ff0000][size=150][br]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. Assume a=2.[br][br][/b] The new function is [b][color=#ff0000]y=2[/color][/b][b][size=150][color=#ff0000][sup]x[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the exponential function.[/color][/b]
Exercise 2
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [/b][b]Assume a=2.[/b] [br][br] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x[/sup][/color][/size][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 3
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. [/b][b]Assume a=2.[/b] [br][br] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x - 3[/sup][/color][/size][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 4
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [/b][b]Assume a=2.[/b] [br][br] The new function is [b][color=#ff0000]y=[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x + 3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 5
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y =[/color] [/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x - 3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 6
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]y =[/color] [/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x + 3[/sup][/color][/size][/b][b][color=#ff0000] - 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 7
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x - 3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x + 3[/sup][/color][/size][/b][b][color=#ff0000] + 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3. [/b][b]Assume a=2.[/b] [br][br] New function: [color=#ff0000] [b]y = 3* [/b][/color][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=3. Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 10
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = (1/3)[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1/3.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 11
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left [br] and a vertical stretch by a factor of 4 . [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = [/color][/b][b][color=#ff0000] 4*[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x+3 [/sup][/color][/size][/b][b][color=#ff0000]+ 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=4.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 12
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left, [br] a vertical shrink by a factor of 1/2 . Assume a =2.[br][br][/b] New function: [b][color=#ff0000]y = (1/2)[/color][/b][b][color=#ff0000]*[/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x+3 [/sup][/color][/size][/b][b][color=#ff0000]+ 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b=1/2.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 13
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Reflection over the x-axis. [/b][b]Assume a=2.[/b] [br][br] New function: [b][color=#ff0000]y = - [/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]x[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=2. Set b = -1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 14
[b][size=150]Perform the following exponential function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [/b][b]Assume a=2.[/b] [br][br] New function: [b] [color=#ff0000]y = [/color][/b][b][color=#ff0000]2[/color][/b][b][size=150][color=#ff0000][sup]-x [/sup][/color][/size][/b], denoted by function f.[br][br] [color=#0000ff] Set a = 2. Set b = 1. [br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]exponential [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 15
[b][br]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].[br] Try using [b][color=#ff0000]0 < a < 1.[/color][/b]
Logarithmic Function Transformations
Logarithmic Function Transformations
Logarithmic Function Transformation Exercise
[b][size=150]The logarithmic function is [color=#ff0000]y = log[sub]a[/sub]x[/color] , denoted by function g. [br][br][/size][/b]The transformed basic function is [b][size=150][color=#ff0000]y = b log[sub]a[/sub](x - h)[/color][/size][/b][b][color=#ff0000] +k where a > 1.[/color][/b][br][b][color=#ff0000][size=150][br]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. Assume a = 2. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]log[sub]2[/sub]x[/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the logarithmic function.[/color][/b]
Exercise 2
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. Assume a = 2.[br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]log[sub]2[/sub]x[/color][/size][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 3
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. Assume a=2.[br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]log[sub]2([/sub]x[/color][/size][/b][b][color=#ff0000] - 3)[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 4
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. Asume a=2.[br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]log[sub]2([/sub]x[/color][/size][/b][b][color=#ff0000] + 3)[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 5
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. Assume a = 2.[br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][size=150][color=#ff0000]log[sub]2([/sub]x[/color][/size][/b][b][color=#ff0000] - 3)[/color][/b][b][color=#ff0000] + 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 6
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. Assume a= 2.[br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][size=150][color=#ff0000]log[sub]2[/sub](x[/color][/size][/b][b][color=#ff0000] + 3)[/color][/b][b][color=#ff0000] - 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 7
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. Assume a=2.[br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][size=150][color=#ff0000]log[sub]2[/sub](x[/color][/size][/b][b][color=#ff0000] - 3)[/color][/b][b][color=#ff0000] - 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 8
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. Assume a =2.[br][br][/b] New function: [b][color=#ff0000]y = [/color][/b][b][size=150][color=#ff0000]log[sub]2[/sub](x[/color][/size][/b][b][color=#ff0000] + 3)[/color][/b][b][color=#ff0000] + 3[/color][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 9
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3. [/b] [b]Assume a =2.[br][/b][br] New function: [color=#ff0000] [b]y = 3 lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub]x[/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 10
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3. [/b][b]Assume a =2.[/b] [br][br] New function: [b][color=#ff0000]y = 1/3 [/color][/b][color=#ff0000][b] lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub]x[/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b][b][color=#ff00ff].[/color][/b]
Exercise 11
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left [br] and a vertical stretch by a factor of 2 . [/b][b]Assume a =2.[/b] [br][br] New function: [b][color=#ff0000]y = 2[/color][/b][color=#ff0000][b] lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub](x + 3 ) + 3[/color][/size][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 12
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up, horizontal shift of 3 units to the left, [br] a vertical shrink by a factor of 1/2 . [/b][b]Assume a =2.[/b] [br] [br] New function: [b][color=#ff0000]y = 1/[/color][/b][b][color=#ff0000]2[/color][/b][color=#ff0000][b] lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub](x + 3 ) + 3[/color][/size][/b], denoted by function f.[br][br] [color=#0000ff]Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which [/color]represents [color=#0000ff]the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 13
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Horizontal stretch by a factor of 1/3. [/b][b]Assume a =2.[/b] [br][br] New function: [color=#ff0000][b]y = [/b][/color][color=#ff0000][b]lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub]((1/3)x)[/color][/size][/b] , denoted by function f.[br][br][color=#0000ff] Place a 1/3 in front of the variable x.[br][/color][color=#0000ff] Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 14
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3. [/b][b]Assume a =2.[/b] [br][br] New function: [b][color=#ff0000]y = [/color][/b][color=#ff0000][b] lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub](2x)[/color][/size][/b], denoted by function f.[br][br][color=#0000ff] Place a 2 in front of the variable x.[br][/color][color=#0000ff] Set b = 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 15
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][br][b]Reflection over the x-axis. [/b][b]Assume a =2.[/b] [br][br] New function: [b][color=#ff0000]y = - [/color][/b][color=#ff0000][b]lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub]x[/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set b = - 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 16
[b][size=150]Perform the following logarithmic function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [/b][b]Assume a =2.[/b] [br][br] New function: [b] [color=#ff0000]y = [/color][/b][color=#ff0000][b]lo[/b][/color][b][size=150][color=#ff0000]g[sub]2[/sub](-x)[/color][/size][/b], denoted by function f.[br][br][color=#0000ff] Place a negative in front of the variable x.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]logarithmic function.[/color][/b]
Exercise 17
[br][b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].
Cube Root Function Transformations
Cube Root Function Transformations
Cube Root Function Transformation Exercise
[b][size=150]The cube function is [color=#ff0000]y = x[/color][color=#ff0000][sup]1/3 [/sup][/color] , denoted by function g. [br][br][/size][/b]The transformed basic function is [b][color=#ff0000]y = a(bx - h)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] +k[/color][/b][br][br][b][color=#ff0000][size=150]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]
Exercise 1
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][color=#ff00ff][b] Observe the transformation of the cubic function.[/b][/color]
Exercise 2
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff].[/color][/b]
Exercise 3
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the right. [br][br][/b] The new function is [b][color=#ff0000]y=(x-3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff]. [/color][/b]
Exercise 4
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=(x+3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]
Exercise 5
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b]y = [color=#ff0000](x-3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 6
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br][/b] New function: [b][color=#ff0000]y = (x+3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 7
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][/b] New function: [b][color=#ff0000]y = (x - 3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 8
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = (x + 3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b],denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 9
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical stretch by a factor of 3. [br][/b] New function: [color=#ff0000] [b]y = 3 [/b][/color][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=3. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]
Exercise 10
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3 [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1/3. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 11
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = (1/3x)[/b][/color][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 12
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3[br][br][/b] New function: [b][color=#ff0000]y = (3x)[/color][/b] [b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 13
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical stretch by a factor of 2. [br][br][/b] New function: [b][color=#ff0000]y = 2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 14
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical shrink by a factor of 1/2. [br][br][/b] New function: [b][color=#ff0000]y = 1/2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]1/3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 15
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]1/3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=-1. Set b = 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]
Exercise 16
[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = (-x)[/color][/b][b][size=150][color=#ff0000][sup]1/3 [/sup][/color][/size][/b], denoted by function f.[br][br] [color=#0000ff] Set a=1. Set b = -1. [br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]
Exercise 17
[br][b]Repeat this exercise as many times as desired until concept is mastered. [br][br][/b] Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].
Finding Real Zeros of Polynomials
Finding Real Zeros of Polynomials
Finding Real Zeros of Polynomials Exercise
[color=#ff0000][b]Practice using the GeoGebra tool above with the following exercises.[br][br][/b][/color][color=#ff00ff][b]Note: You can zoom in or out with the mouse.[/b][/color]
Exercise 1
[b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][br][/color][/b][size=150][i][b][color=#ff00ff][size=200][size=100]f (x) = x[sup]3 [/sup]- 4x[sup]2[/sup][/size][size=100][sup][/sup] -4x +16[/size][/size][/color][/b][/i][/size]
Exercise 2
[b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][/color][/b][i][color=#ff00ff][b][size=150][size=200][size=100]f (x) = 4x[sup]5 [/sup]- 8x[sup]4 [/sup]- 5x[sup]3[/sup] +10 x[sup]2[/sup]+ x - 2[/size][/size][/size][/b][/color][/i]
Exercise 3
[size=150][b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][/color][/b][i][b][color=#ff00ff][size=200][size=100]f (x) = 3x[sup]3 [/sup]+2x[sup]2[/sup] -37x +12[/size][/size][/color][/b][/i][/size]
Exercise 4
[size=150][b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][/color][/b][i][b][color=#ff00ff][size=200][size=100]f (x) = 3x[sup]3 [/sup]+2x[sup]2[/sup] -61x +20[br][/size][/size][/color][/b][/i][/size]
Exercise 5
[size=150][b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][/color][/b][i][b][color=#ff00ff][size=200][size=100]f (x) = x[sup]3 [/sup]- 13x -12[/size][/size][/color][/b][/i][/size]
Exercise 6
[size=150][b][color=#9900ff]Enter the following polynomial [/color][/b][b][color=#9900ff]in the input box above [/color][/b][b][color=#9900ff]and find the real zeros: [br][/color][/b][i][b][color=#ff00ff]f (x) = 4x[sup]4 [/sup]- 99 x[sup]2[/sup] -65x +300[/color][/b][/i][/size]