Chapter 1
Functions and Their Graphs
Warm Up 1.4
 A Norman window has the shape of a square with
a semicircle mounted on it. Find the width of the
window if the total area of the square and the
semicircle is to be 200 ft2.
x
x
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1.4 Transformation of
Functions
 Objectives:
Recognize graphs of common functions.
Use vertical and horizontal shifts and
reflections to graph functions.
Use nonrigid transformations to graph
functions.
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Vocabulary
 Constant Function
 Identity Function
 Absolute Value Function
 Square Root Function
 Quadratic Function
 Cubic Function
 Transformations of Graphs
 Vertical and Horizontal Shifts
 Reflection
 Vertical and Horizontal Stretches & Shrinks
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Common Functions

Sketch graphs of the following functions:
1.
Constant Function
2.
Identity Function
3.
Absolute Value Function
4.
Square Root Function
5.
Quadratic Function
6.
Cubic Function
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Constant Function
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Identity Function
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Absolute Value Function
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Square Root Function
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Quadratic Function
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Cubic Function
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Exploration 1
 Graph the following functions in the same viewing
window:
y = x2 + c, where c = –2, 0, 2, and 4.
Describe the effect that c has on the graph.
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Exploration 2
 Graph the following functions in the same viewing
window:
y = (x + c)2, where c = –2, 0, 2, and 4.
Describe the effect that c has on the graph.
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Vertical and Horizontal Shifts

Let c be a positive real number.
Shifts in the graph of y
= f (x) are as follows:
1. h(x) = f (x) + c
______________________
2. h(x) = f (x) – c
______________________
3. h(x) = f (x – c)
______________________
4. h(x) = f (x + c)
______________________
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Example 1
Compare the graph of each function with the graph
of f (x) = x3 without using your graphing calculator.
a.
g(x) = x3 – 1
b.
h(x) = (x – 1)3
c.
k(x) = (x + 2)3 + 1
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Example 2
 Use the graph of f (x) = x2
to find an equation for
g(x) and h(x).
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Exploration 3

Compare the graph of each function with the graph
of f (x) = x2 by using your graphing calculator to
graph the function and f in the same viewing
window. Describe the transformation.
a.
g(x) = –x2
b.
h(x) = (–x)2
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Reflections in the Coordinate Axes

Reflections in the coordinate axes of the graph of
y = f (x) are represented as follows:
1.
h(x) = –f (x) _______________________
2.
h(x) = f (–x)
_______________________
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Example 3
 Use the graph of f (x) = x4
to find an equation for
g(x) and h(x).
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Example 4
Compare the graph of each function with the graph of
f ( x)  x .
a. g ( x)   x
b. h( x)   x
c. k ( x)   x  2
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Exploration 4
 Graph the following functions in the same viewing
window:
y = cx3, where c = 1, 4 and ¼.
Describe the effect that c has on the graph.
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Exploration 5
 Graphing the following functions in the same viewing
window:
y = (cx)3, where c = 1, 4 and ¼.
Describe the effect that c has on the graph.
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Nonrigid Transformations
 Rigid Transformation
Changes position of the graph but maintains the shape
of the original function.
Horizontal or vertical shifts and reflections.
 Nonrigid Transformation
Causes a distortion in the graph.
Changes the shape of the original graph.
Vertical or horizontal stretches and shrinks.
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Vertical Stretch or Shrink
 Original function y = f (x).
 Transformation y = c f (x).
Each y-value is multiplied by c.
Vertical stretch if c > 1.
Vertical shrink if 0 < c < 1.
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Horizontal Stretch or Shrink
 Original function y = f (x).
 Transformation y = f (cx).
Each x-value is multiplied by 1/c.
Horizontal shrink if c > 1.
Horizontal stretch if 0 < c < 1.
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Homework 1.4
 Worksheet 1.4
#5, 7, 11, 13, 16, 20, 24, 26, 27, 33, 37, 39, 42, 45, 47, 51,
53, 57, 61, 63, 67
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