Function Families and Transformations
An investigation into some function families: rational, exponential, logarithmic, quadratic, and trigonometric functions, including their transformations.
Table des matières
- Rational Functions
- Reciprocal Function
- Rational Functions
- Exponential and Logarithmic Functions
- Graphs of Exponential Functions
- Transformations
- Translations Part 1
- Translations Part 2
- Streches Part 1
- Streches Part 2
- Reflections
- Mixed Transformations - Vertical
- Mixed Transformations - Horizontal
- Trigonometric Functions
- Sine Transformations
- Cosine Curve
- Tangent Curve
- Quadratics
- Quadratic Functions Part 1
- Quadratic Functions Part 2
- Quadratic Functions Part 3
- Polynomials
- Graphing Cubic Equations
Reciprocal Function
The graph below is the reciprocal function. You can change k, and notice the effects.[br][br]Answer the questions below, in your notebook.
[b]Question 1:[/b] Why is this function called the "reciprocal" function? What does reciprocal mean in mathematics?[br][br][b]Question 2: [/b]Make a list of the properties of the reciprocal function, when [math]k=1[/math]. [br][b]Hint: [/b]Discuss overall shape, symmetry, asymptotes, ect.[br][br][b]Question 3: [/b]Switch [math]k[/math] from positive to negative. What changes? Describe the reciprocal function, when [math]k[/math] is negative.[br][br][b]Question 4: [/b]Consider [math]\left|k\right|[/math], the absolute value of k. Describe how the function changes when [math]\left|k\right|[/math]increases and decreases. Check this for both positive and negative values of [math]k[/math].[b][br][/b]
Graphs of Exponential Functions
This application will help us visualize the transformations of exponential functions.[br][br]On the right, you can cycle through questions, answering them in your notebook. For each question, a slider is shown, to allow you to change some values of the graph.
Translations Part 1
This applet has 5 different functions. Your goals, will be to see how the function f(x) is transformed, when going to f(x)+d.[br][br]You can move slider "d", and push "Switch Function" to cycle through the 5 available functions. Try it out, and then answer the questions below.
This type of transformation, is called [b]translations[/b]. [br][br][b]Question 1: [/b] Using the word [b]translation[/b], describe the effect of "d", when [b]d>0[/b], when y=f(x) is transformed to y=f(x)+d. [b]Be specific![br][br][/b][b]Question 2: [/b]Using the word [b]translation[/b], describe the effect of "d", when [b]d<0[/b], when y=f(x) is transformed to y=f(x)+d. Be specific!
Sine Transformations
The applet below has 4 different transformations of the curve, y=sin x. You can cycle through them, by pushing "Next Transformation", and move the sliders for a,b,c, and d. Note: x is in radians.[br][br]Play around until you understand the idea, and then answer the questions below.
For all questions, consider the points [b]minimum, maximum, amplitude, principal axis, period[/b].[b][br][br]Question 1: (a sin x) [br][/b]a. Change a. What changes, and what stays the same?[br]b. How does a affect the function y=a sin x?[br]c. Choose three different a values, and write down the minimum, maximum, and amplitude.[br]d. Hence, what is the amplitude of y=3.56sin x? y=-4sin x? y=a sin x?[br][br][b]Question 2: (sin (bx)) [br][/b]a. Change b. What changes, and what stays the same?[br]b. How does b affect the function y=sin (bx)?[br]c. Choose three different b values, and write down the periods.[br]d. Hence, what is the period of y=sin (3x)? y=sin (1/2 x)? y=sin(bx)?[br][br][b]Question 3: (sin (x-c)) [br][/b]a. Change c. What changes, and what stays the same?[br]b. How does c affect the function y=sin (x-c)?[br][br][b]Question 4: (sin x + d) [br][/b]a. Change d. What changes, and what stays the same?[br]b. How does d affect the function y=sin x + d?[br]c. Choose three different d values, and write down the equations of the principal axis.[br]d. Hence, what is the principle axis of y=sin x + 5? y=sin x - 3? y=sin x + d?[br][br][b]Question 5: (General Case)[br][/b]From previous questions, find the minimum, maximum, period, principal axis, and amplitude, of the function [br]y=a sin(b(x-c))+d.
Quadratic Functions Part 1
The applet below has the graph of y=x^2, as well as y=ax^2. You can change the value of a, by moving the slider.[br][br]Move the slider a, and see how the graph changes. Then, answer the questions below in your notebook.
[b]Question 1: [/b]How does "a" affect how the graph of y=ax^2 "opens"?[br][br][b]Question 2: [/b]How does "a" affect the shape of the graph? Be careful about positive and negative values for "a"!
Graphing Cubic Equations
The applet below has 4 different cubic functions, including their roots. You can cycle through the functions, as well as change some values of the functions. Play around with the app, and then answer the questions below in your notebook.
[b]Question 1: [/b]Consider the function f_1(x), which has three real, distinct zeros: b, c, and d, and a leading coefficient a. How does "a" affect the graph?[br][br]What is the geometrical significance of b, c, and d?[br][br]Draw some examples, illustrating the above answers.[br][br][b]Question 2: [/b]Consider the function f_2(x), which has two real zeroes, with one repeated, and a leading coefficient a. What is the geometrical significance of the squared factor?[br][br]Draw an example, illustrating the above answers.[br][br][b]Question 3:[/b] Consider the function f_3(x), which has one zero, repeated three times, and a leading coefficient a. What is the geometrical significance of the cubed factor?[br][br]Draw an example, illustrating the above answer.[br][br][b]Question 4: [/b]Consider the function f_4(x), which has one real zero, and two complex zeros (so an irreducible quadratic, with a negative discriminant). Compare this function to the previous function, f_3(x). [br][br]Draw an example, illustrating the above answer.