Consider the following parent function



Another Look at Scale Changes(Answers)

When graphing a translated function, most people focus on moving the standard parent function up, down, left, or right of the standard coordinate axes. In the activity “Another Look at Translations”, the standard axes were translated first and then the standard parent function was drawn at the new location about the translated axes. The initial focus was on the axes not the graph. When graphing a function affected by scale changes, most people focus on stretching or compressing the graph of the function relative away from or towards the standard coordinate axes. In this activity, you will learn a strategy for graphing functions that are affected by scale changes that will focus on the coordinate axes first and the parent function second.

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Consider the parent function [pic] and the related transformed functions drawn in the window: [-4.7,4.7] x [-8,8] with an Xscl of 1 and a Yscl of 1.

|Parent: |Transformed: |Transformed: |Transformed: |Transformed: |

|[pic] |[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |[pic] |

|x-intercepts |x-intercepts |x-intercepts |x-intercepts |x-intercepts |

|(-2,0)&(2,0) |(-1,0)&(1,0) |(-4,0)&(4,0) |(-2,0)&(2,0) |(-2,0)&(2,0) |

|y-intercept |y-intercept |y-intercept |y-intercept |y-intercept |

|(0,-4) |(0,-4) |(0,-4) |(0,-8) |(0,-2) |

Each of the transformed functions maintains the general shape of the parent function but is either compressed towards or stretched away from one of the axes. Graphing each parent function and their related transformed functions in the same window allows us to visually see the compressions and stretches. Focusing on the intercepts of the quadratic function and the period and amplitude of the sine function allows us to quantify the magnitude of the scale changes. Remember that compression has a scale factor between 0 and 1.

Question 1: Using the information above and what you already know about scale changes complete the following chart:

|Function: |Horizontal(H) or Vertical(V) |Compression(C) or Stretch (S)|Scale Factor |Mapping: |

| |Scale Change | | |(x,y)[pic](?,?) |

|[pic] | | | | |

| |H |C |.5 |(.5x,y) |

|[pic] |(H) |(S) |(2) |(2x,y) |

|[pic] |(V) |(S) |(2) |(x,2y) |

|[pic] |(V) |(C) |(.5) |(x,.5y) |

Question 2: Graphing each parent function and their related transformed functions in the same window allowed us to visually see the compressions and stretches. In the following tables, determine the window that was used to create each graph so that each transformed function “appears” to be identical to the original parent function. In other words, change the scales so that the graphs appear congruent. One problem in each table has been completed for you.

|Parent: |Transformed: |Transformed: |Transformed: |Transformed: |

|[pic] |[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |[pic] |

|x-axis |x-axis |x-axis |x-axis |x-axis |

| | | | | |

|Xmin = -4.7 |Xmin = -2.35 |Xmin = (-9.4) |Xmin = (-4.7) |Xmin = (-4.7) |

| | | | | |

| | | | | |

|Xmax = 4.7 |Xmax = 2.35 |Xmax = (9.4) |Xmax = (4.7) |Xmax = (4.7) |

| | | | | |

| | | | | |

|Xscl = 1 |Xscl = .5 |Xscl = (2) |Xscl = (1) |Xscl = (1) |

| | | | | |

|y-axis |y-axis |y-axis |y-axis |y-axis |

| | | | | |

|Ymin = -8 |Ymin = -8 |Ymin = (-8) |Ymin = (-16) |Ymin = (-4) |

| | | | | |

| | | | | |

|Ymax = 8 |Ymax = 8 |Ymax = (8) |Ymax = (16) |Ymax = (4) |

| | | | | |

| | | | | |

|Yscl = 1 |Yscl = 1 |Yscl = (1) |Yscl = (2) |Yscl = (-.5) |

| | | | | |

Question 3: Discuss any relationships that you see between the new windows that you found in Question 2 and the horizontal and vertical scale changes that you found in Question 1. In other words, describe exactly how to determine the new “scales” once the horizontal and vertical scale changes are identified.

(All you have to do is to determine the new window is to determine the horizontal and vertical scale factors and apply those scale factors to the appropriate part of the window. For example, when the horizontal scale factor is .5, the original Xmin, Xmax, and Xscl were all multiplied by .5, and since there was no vertical scale change the Ymin,Ymax,and Yscl remained the same)

Question 4: See if the patterns that you discovered in Question 1 can be applied to other functions in other windows. Complete the following charts and check your answers with your graphing calculator.

|Parent Function [pic] |Transformed Function [pic] |

|Graph of Parent Function |Window for Parent |Verbal Description |Window for Transformed Function |

| |Function |Of Transformation | |

| |[-4.7,4.7] |Horizontal Scale Factor: .25 |([-1.175,1.175] |

|[pic] |x | |x |

| |[-3.1,3.1] | |[-6.1,6.1] |

| | | | |

| |Xscl: 1 | |Xscl: .25 |

| | | | |

| |Yscl: 1 | |Yscl: 2) |

| | |Vertical Scale Factor: 2 | |

|Parent Function [pic] |Transformed Function [pic] |

|Graph of Parent Function |Window for Parent |Verbal Description |Window for Transformed Function |

| |Function |Of Transformation | |

| |[-180,180] |Horizontal Scale Factor: (.2) | |

|[pic] |x | |([-36,36] |

| |[-10,10] | |x |

| | | |[-10,10] |

| |Xscl: 45 | | |

| | | |Xscl=9 |

| |Yscl: 2 | |Yscl=2] |

| | |Vertical Scale Factor: (1) | |

|Parent Function y=int(x) |Transformed Function y=2int(.25x) |

|Graph of Parent Function |Window for Parent |Verbal Description |Window for Transformed Function |

| |Function |Of Transformation | |

| |[-9.4,9.4] |Horizontal Scale Factor: (4) | |

|[pic] |x | |([-37.6,37.6] |

| |[-10, 10] | |x |

| | | |[-20,20] |

| |Xscl: 1 | | |

| | | |Xscl =4 |

| |Yscl: 1 | |Yscl = 2 |

| | |Vertical Scale Factor: (2) | |

|Parent Function [pic] |Transformed Function [pic] |

|Graph of Parent Function |Window for Parent |Verbal Description |Window for Transformed Function |

| |Function |Of Transformation | |

| |[-4.7,4.7] |Horizontal Scale Factor: (1) |[-4.7,4.7] |

|[pic] |x | |x |

| |[-3.1,3.1] | |[-9.3,9.3] |

| | | | |

| |Xscl: 1 | |Xscl: 1 |

| | | | |

| |Yscl: 1 | |Yscl: 3 |

| | |Vertical Scale Factor: (3) | |

Question 5: Consider the following strategy for graphing functions affected by scale changes. The concept of “scale” change is being taken literally here. Note that it is again based on transforming the axes first and then graphing the standard parent function second.

1. Graph the parent function in some “nice” window, clearly labeling each axis.

2. Identify the horizontal and vertical scale factors.

3. Re-draw the parent function on an unlabeled graph grid

4. Use your knowledge of the vertical and horizontal scale factors to label the new axes.

Do you believe that the strategy is valid? Justify your answer.

(yes, changing the scale worked in the previous question and the graph of the transformed function looked exactly the same as the parent function in the original window)

Question 6: Give the new strategy a try. Complete the following table.

|Transformed and Parent |Graph of Parent |Verbal Description of |Graph of Transformed Function |

|Functions |Function |Transformation | |

| | | | |

|[pic] |[pic] |Hor. Scale Factor: .5 |[pic] |

| | | | |

|y=cos(x) | |Vert. Scale | |

| | |Factor: 3 | |

|Note that each labeled number on the new horizontal axis is half the value of the corresponding number on the original horizontal axis and that each |

|labeled number on the new vertical axis is 3 times the corresponding number on the original vertical axis. |

|y=.5sin(.25x) | | | |

| |[pic] |Hor. Scale Factor: (4) |[pic] |

| | | | |

| | |Vert. Scale | |

| | |Factor: (.5) | |

|[pic] |[pic] | |[pic] |

| | | | |

| | |Hor. Scale Factor: (2) | |

| | | | |

| | |Vert. Scale | |

| | |Factor: (4) | |

|[pic] |[pic] | |[pic] |

| | |Hor. Scale Factor: (.25) | |

| | | | |

| | |Vert. Scale | |

| | |Factor: (25) | |

| | | |[pic] |

|[pic] |[pic] |Hor. Scale Factor: (.2) | |

| | | | |

| | |Vert. Scale | |

| | |Factor: (10) | |

Question 7: Did you find it easy or hard to complete the table and graph the functions? State advantages or disadvantages that this method of transforming the axes first and then drawing the standard parent function has over the more traditional method of using a standard coordinate system and transforming the graph.

(Easy, for each pair of functions, the graphs looked the same, the scales on the axes changed. Advantage: works for very large or small scale factors, will help me find the proper windows to graph on calculator. Disadvantage: don’t visually see the stretches or compressions)

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