Function Transformation Lesson

Discover the Stretch, Compression ,Horizontal and Vertical shifts

You may try to Change the numbers which cause different effects of the functions

First Enter your main Function and then change the values to see the transformed function

Notes-1

Use the Applet above and check this:[br][br]We can stretch or compress f(x) in the y-direction by multiplying the whole function by a constant.[br][br][br][math]f\left(x\right)=x^2[/math] [math]g\left(x\right)=C.\left(x^2\right)[/math] [br] if C> 1 : it will stretch the f(x) in the y-direction [br]if 0 < C < 1: it will compress the function in the y-direction.[br][br]go up and try[br]

Notes-2

Use the Applet above and check this:[br][br]We can stretch or compress f(x) in the x-direction by multiplying 'x' by a constant.[br][br][br][math]f\left(x\right)=x^2[/math] [math]g\left(x\right)=\left(D.x\right)^2[/math] [br] [list][*]D > 1 compresses it[/*][*]0 < D < 1 stretches it[/*][/list]Note that (unlike for the y-direction), [b]bigger[/b] values cause more [b]compression[/b].[br]

Using Notes 1 & 2 answer the following questions

Q1. Compare g(x) with f(x)[br][math]f\left(x\right)=x^2[/math][br][math]g\left(x\right)=\frac{1}{4}x^2[/math]

[math]\frac{1}{4}[/math] will cause [color=#0000ff][b]compression in x-direction[/b][/color] [math]\frac{1}{4}[/math]will cause compression in [b][color=#0000ff]y-direction[/color][/b] [math]\frac{1}{4}[/math]will cause [b][color=#0000ff]stretch in y-direction[/color][/b] [math]\frac{1}{4}[/math]will cause [b][color=#0000ff]stretch in x-direction[/color][/b]

Q2.

Compare g(x) with f(x)[br][math]f\left(x\right)=x^2[/math][br][math]g\left(x\right)=\left(\frac{x}{3}\right)^2[/math]

[math]\frac{1}{3}[/math]will cause [b][color=#0000ff]compression in y-direction[/color][/b] [math]\frac{1}{3}[/math]will cause [b][color=#0000ff]stretch in x-direction[/color][/b] [math]\frac{1}{3}[/math] will cause [color=#0000ff][b]compression in x-direction[/b][/color] [math]\frac{1}{3}[/math]will cause [b][color=#0000ff]stretch in y-direction[/color][/b]

Q3.

Compare g(x) with f(x)[br][math]f\left(x\right)=x^2[/math][br][math]g\left(x\right)=2\left(\frac{x}{3}\right)^2[/math]

[math]\frac{1}{3}[/math] and 2 both will cause [color=#0000ff][b]compression in x-direction, [/b][/color] [math]\frac{1}{3}[/math]will cause [b][color=#0000ff]stretch in x-direction, 2 will stretch in y direction[/color][/b] [math]\frac{1}{3}[/math]and 2 both will cause [b][color=#0000ff]compression in y-direction[/color][/b] [math]\frac{1}{3}[/math]will cause [b][color=#0000ff]stretch in x-direction, 2 will compress in y.direction[/color][/b]

Notes-3

We can move it [b]up or down by adding a constant[/b] to the [color=#0000ff]y-value[/color][br]g(x) = x[sup]2[/sup] + K or [math]g\left(x\right)=f\left(x\right)+K[/math][br]Note: to move the line [b]down[/b], we use a [b]negative[/b] value for K.[br][list][*]K > 0 [color=#dd7e6b][b]moves it up[/b][/color][/*][*]K < 0 [color=#93c47d][b]moves it down[/b][/color][/*][/list]

Notes-4

We can move it [b]left or right by adding a constant[/b] to the [color=#0000ff]x-value[/color][br]g(x) = (x+h)[sup]2[/sup] or [math]g\left(x\right)=f\left(x+h\right)[/math][br]Note: [b]Adding h[/b] moves the function to the [b]left[/b] (the negative direction).[br][list][*]h > 0 [color=#dd7e6b][b]moves it left[/b][/color][/*][*]h < 0 [color=#93c47d][b]moves it right[/b][/color][/*][/list][math]\left(x-3\right)^2[/math] will move 3 units to the right of the y-axis , beause x-3= 0 or x= 3 (positive menas right)[br][br][math]\left(x+4\right)^2[/math] will move 4 units to the left of y-axis because x+4= 0 means x= -4 ( so left)

Based on Notes 3 and 4 answer these Questions

Q4.[br]if [br][math]f\left(x\right)=x^2[/math] and [br][math]g\left(x\right)=\left(x+1\right)^2-3[/math][br][br]compare g(x) with f(x)

+1 will move the function [b][color=#0000ff]to the left[/color][/b] and -3 will move [b][color=#6aa84f]to down[/color][/b] +1 will move the function [b][color=#0000ff]to the right[/color][/b] and -3 will move [b][color=#6aa84f]to up[/color][/b] +1 will move the function [b][color=#0000ff]to the right[/color][/b] and -3 will move [b][color=#6aa84f]to the left[/color][/b] +1 will move the function [b][color=#0000ff]to the left[/color][/b] and -3 will move [b][color=#6aa84f]to the right[/color][/b]

Q5.

if [br][math]f\left(x\right)=x^2[/math] and [br][math]g\left(x\right)=\left(x-2\right)^2+5[/math][br][br]compare g(x) with f(x)

-2 will move the function [b][color=#0000ff]to the right[/color][/b] and +5 will move [b][color=#6aa84f]to down[/color][/b] -2 will move the function [b][color=#0000ff]to the right[/color][/b] and +5 will move [b][color=#6aa84f]to up[/color][/b] -2 will move the function [b][color=#0000ff]to the left[/color][/b] and +5 will move [b][color=#6aa84f]to the right[/color][/b] -2 will move the function [b][color=#0000ff]to the left[/color][/b] and +5 will move [b][color=#6aa84f]to down[/color][/b]

Q6. Use all the facts you have learned

if [br][math]f\left(x\right)=x^2[/math] and [br][math]g\left(x\right)=2\left(x-3\right)^2+1[/math][br][br]compare g(x) with f(x)

2 will [b][color=#cc4125]stretch in the x-direction[/color][/b], -3 move the function [b][color=#0000ff]to the left[/color][/b] and +1 will move [b][color=#6aa84f]to Up[/color][/b] 2 will compress[b][color=#cc4125] in the x-direction[/color][/b], -3 move the function [b][color=#0000ff]to the right[/color][/b] and +1 will move [b][color=#6aa84f]to down[/color][/b] 2 will [b][color=#cc4125]compresses in the y-direction[/color][/b], -3 move the function [b][color=#0000ff]to the left[/color][/b] and +1 will move [b][color=#6aa84f]to the right[/color][/b] 2 will [b][color=#cc4125]stretch in the y-direction[/color][/b], -3 move the function [b][color=#0000ff]to the right[/color][/b] and +1 will move [b][color=#6aa84f]to Up[/color][/b]

Q7. What would be g(x) if

we transform f(x)= x[sup]2 [br][/sup]such that it [b][color=#dd7e6b]compresses by a factor of 3 in the x-direction[/color][/b] , shift the function by [b][color=#0000ff]2 on the left of y-axis[/color][/b] and [b][color=#6aa84f]move 4 units to Up[/color][/b]