Project AMP Dr. Antonio R. Quesada Director Project AMP

Another Look at Scale Changes

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 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:

x-intercepts
(-2,0)&(2,0) / x-intercepts
(-1,0)&(1,0) / x-intercepts
(-4,0)&(4,0) / x-intercepts
(-2,0)&(2,0) / x-intercepts
(-2,0)&(2,0)
y-intercept
(0,-4) / y-intercept
(0,-4) / y-intercept
(0,-4) / y-intercept
(0,-8) / y-intercept
(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)
Scale Change / Compression(C) or Stretch (S) / Scale Factor / Mapping:
(x,y)(?,?)
/ H /

C

/ .5 / (.5x,y)

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:

x-axis
Xmin = -4.7
Xmax = 4.7
Xscl = 1 / x-axis
Xmin = -2.35
Xmax = 2.35
Xscl = .5 / x-axis
Xmin =
Xmax =
Xscl = / x-axis
Xmin =
Xmax =
Xscl = / x-axis
Xmin =
Xmax =
Xscl =
y-axis
Ymin = -8
Ymax = 8
Yscl = 1 / y-axis
Ymin = -8
Ymax = 8
Yscl = 1 / y-axis
Ymin =
Ymax =
Yscl = / y-axis
Ymin =
Ymax =
Yscl = / y-axis
Ymin =
Ymax =
Yscl =

Question 3: Discuss any relationships that you see between the new windows that you found in Question2 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.

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 / Transformed Function
Graph of Parent Function / Window for Parent
Function / Verbal Description
Of Transformation / Window for Transformed Function
/ [-4.7,4.7]
x
[-3.1,3.1]
Xscl: 1
Yscl: 1 / Horizontal Scale Factor: .25
Vertical Scale Factor: 2
Parent Function / Transformed Function
Graph of Parent Function / Window for Parent
Function / Verbal Description
Of Transformation / Window for Transformed Function
/ [-180,180]
x
[-10,10]
Xscl: 45
Yscl: 2 / Horizontal Scale Factor:
Vertical Scale Factor:
Parent Function y=int(x) / Transformed Function y=2int(.25x)
Graph of Parent Function / Window for Parent
Function / Verbal Description
Of Transformation / Window for Transformed Function
/ [-9.4,9.4]
x
[-10, 10]
Xscl: 1
Yscl: 1 / Horizontal Scale Factor:
Vertical Scale Factor:
Parent Function / Transformed Function
Graph of Parent Function / Window for Parent
Function / Verbal Description
Of Transformation / Window for Transformed Function
/ [-4.7,4.7]
x
[-3.1,3.1]
Xscl: 1
Yscl: 1 / Horizontal Scale Factor:
Vertical Scale Factor:

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.

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

Transformed and Parent Functions / Graph of Parent
Function / Verbal Description of Transformation / Graph of Transformed Function

y=cos(x) / / Hor. Scale Factor: .5
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) / / Hor. Scale Factor:
Vert. Scale
Factor: /
/ / Hor. Scale Factor:
Vert. Scale
Factor: /
/ / Hor. Scale Factor:
Vert. Scale
Factor: /
/ / Hor. Scale Factor:
Vert. Scale
Factor: /

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.