Functions and Their Graphs Chapter 1 Overview

Functions and Their Graphs Chapter 1 Overview

Functions and Their Graphs explores the foundational concepts of functions, including their definitions, types, and graphical representations. This chapter emphasizes the importance of understanding parent functions and transformations such as shifts, reflections, and stretches. Students will learn how to manipulate graphs through vertical and horizontal shifts, as well as how to apply these transformations to sketch various functions. Ideal for high school mathematics students, this chapter serves as a critical resource for mastering algebraic concepts and preparing for advanced topics in calculus and analytical geometry.

Key Points

  • Explains the concept of parent functions and their transformations.
  • Covers vertical and horizontal shifts of function graphs.
  • Details reflections in the coordinate axes and their effects on graphs.
  • Includes examples and exercises for practicing graph transformations.
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74 Chapter 1 Functions and Their Graphs
Shifting Graphs
Many functions have graphs that are simple transformations of the parent graphs
summarized in Section 1.6. For example, you can obtain the graph of
by shifting the graph of upward two units, as shown in Figure 1.76. In
function notation, and are related as follows.
Upward shift of two units
Similarly, you can obtain the graph of
by shifting the graph of to the right two units, as shown in Figure 1.77.
In this case, the functions and have the following relationship.
Right shift of two units
FIGURE 1.76 FIGURE 1.77
The following list summarizes this discussion about horizontal and vertical
shifts.
x
−112
2
1
3
3
4
g(x) = (x − 2)
2
f(x) = x
2
y
x
−2 −112
1
3
4
h(x) = x
2
+ 2
f(x) = x
2
y
f
x 2
g
x
x 2
2
fg
f
x
x
2
g
x
x 2
2
f
x
2h
x
x
2
2
fh
f
x
x
2
h
x
x
2
2
What you should learn
•Use vertical and horizontal
shifts to sketch graphs of
functions.
•Use reflections to sketch
graphs of functions.
•Use nonrigid transformations
to sketch graphs of functions.
Why you should learn it
Knowing the graphs of common
functions and knowing how
to shift, reflect, and stretch
graphs of functions can help
you sketch a wide variety of
simple functions by hand.This
skill is useful in sketching graphs
of functions that model real-life
data, such as in Exercise 68 on
page 83, where you are asked to
sketch the graph of a function
that models the amounts of
mortgage debt outstanding
from 1990 through 2002.
Transformations of Functions
©Ken Fisher/Getty Images
1.7
Vertical and Horizontal Shifts
Let be a positive real number. Vertical and horizontal shifts in the graph
of are represented as follows.
1. Vertical shift units upward:
2. Vertical shift units downward:
3. Horizontal shift units to the right:
4. Horizontal shift units to the left: h
x
f
x c
c
h
x
f
x c
c
h
x
f
x
cc
h
x
f
x
cc
y f
x
c
In items 3 and 4, be sure you
see that
corresponds to a right shift and
corresponds to
a left shift for
c
>
0.
h
x
f
x c
h
x
f
x c
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Section 1.7 Transformations of Functions 75
Some graphs can be obtained from combinations of vertical and horizontal
shifts, as demonstrated in Example 1(b). Vertical and horizontal shifts generate a
family of functions, each with the same shape but at different locations in the
plane.
Shifts in the Graphs of a Function
Use the graph of to sketch the graph of each function.
a.
b.
Solution
a. Relative to the graph of the graph of is a downward
shift of one unit, as shown in Figure 1.78.
b. Relative to the graph of the graph of involves
a left shift of two units and an upward shift of one unit, as shown in Figure
1.79.
FIGURE 1.78 FIGURE 1.79
Now try Exercise 1.
In Figure 1.79, notice that the same result is obtained if the vertical shift
precedes the horizontal shift or if the horizontal shift precedes the vertical shift.
x
h(x) = (x + 2)
+ 1
3
3
f(x) = x
y
−4 −2
−3
−2
−1
1
2
3
−112
x
−21
2
−2
2
gx x() = 1−
3
1
−1
fx x() =
3
y
h
x
x 2
3
1f
x
x
3
,
g
x
x
3
1f
x
x
3
,
h
x
x 2
3
1
g
x
x
3
1
f
x
x
3
You might also wish to illustrate simple
transformations of functions numerically
using tables to emphasize what happens
to individual ordered pairs. For instance,
if you have
,and
you can illustrate these transformations
with the following tables.
4
1
00
11
24
04
11
20
31
44
4 2 2
3 2 1
2 2 0
1 2 1
0 2 2
g
x
f
x 2
x 2x
4 2 6
1 2 3
0 2 2
1 2 31
4 2 62
h
x
f
x
2f
x
x
g
x
x 2
2
f
x 2
,f
x
2
f
x
x
2
, h
x
x
2
2
Graphing utilities are ideal tools for exploring translations of functions.
Graph and in same viewing window. Before looking at the graphs,
try to predict how the graphs of and relate to the graph of
a.
b.
c. f
x
x
2
, g
x
x 4
2
, h
x
x 4
2
2
f
x
x
2
, g
x
x 1
2
, h
x
x 1
2
2
f
x
x
2
, g
x
x 4
2
, h
x
x 4
2
3
f.hg
hf, g,
Exploration
Example 1
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76 Chapter 1 Functions and Their Graphs
Reflecting Graphs
The second common type of transformation is a reflection. For instance, if you
consider the -axis to be a mirror, the graph of
is the mirror image (or reflection) of the graph of
as shown in Figure 1.80.
Finding Equations from Graphs
The graph of the function given by
is shown in Figure 1.81. Each of the graphs in Figure 1.82 is a transformation of
the graph of Find an equation for each of these functions.
(a) (b)
FIGURE 1.82
Solution
a. The graph of is a reflection in the -axis followed by an upward shift of two
units of the graph of So, the equation for is
b. The graph of is a horizontal shift of three units to the right followed by a
reflection in the -axis of the graph of So, the equation for is
Now try Exercise 9.
h
x

x 3
4
.
hf
x
x
4
.x
h
g
x
x
4
2.
gf
x
x
4
.
xg
−1
5
−3
1
yhx=()
−3
3
−1
ygx=()
3
f.
f
x
x
4
f
x
x
2
,
h
x
x
2
x
Reflections in the Coordinate Axes
Reflections in the coordinate axes of the graph of are represented
as follows.
1. Reflection in the -axis:
2. Reflection in the -axis: h
x
f
x
y
h
x
f
x
x
y f
x
x
−2
−2
−112
1
2
−1
f (x) = x
2
2
h(x) = −x
y
FIGURE 1.80
−3
3
3
−1
fx x() =
4
FIGURE 1.81
Reverse the order of transfor-
mations in Example 2(a). Do
you obtain the same graph?
Do the same for Example 2(b).
Do you obtain the same graph?
Explain.
Exploration
Example 2
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End of Document
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FAQs of Functions and Their Graphs Chapter 1 Overview

What are parent functions and why are they important?
Parent functions are the simplest forms of functions that serve as the foundation for more complex functions. Understanding parent functions is crucial because they provide a baseline for graph transformations, allowing students to predict how changes in equations affect the shape and position of graphs. For example, the parent function of a quadratic is f(x) = x², and transformations like shifts or reflections can be applied to this basic shape to create a variety of other quadratic functions.
How do vertical and horizontal shifts affect the graph of a function?
Vertical shifts move the graph of a function up or down without changing its shape, while horizontal shifts move it left or right. For instance, if a function f(x) is shifted vertically by k units, the new function becomes f(x) + k. Similarly, a horizontal shift to the right by h units transforms f(x) into f(x - h). These shifts help in understanding how functions behave in different contexts and are essential for graphing functions accurately.
What is the significance of reflections in function graphs?
Reflections in function graphs involve flipping the graph over a specific axis, which changes the orientation of the function. For example, reflecting a function over the x-axis transforms f(x) into -f(x), while reflecting over the y-axis changes it to f(-x). These transformations are important for understanding symmetry in functions and can help identify properties such as evenness and oddness, which are critical in calculus and algebra.
What types of transformations can be applied to functions?
Transformations that can be applied to functions include vertical shifts, horizontal shifts, reflections, and nonrigid transformations such as stretches and shrinks. Each type of transformation alters the graph's position or shape in a specific way. For example, a vertical stretch multiplies the y-values of a function by a factor greater than one, while a vertical shrink does the opposite. Understanding these transformations allows students to manipulate and analyze functions effectively.
How can students practice graph transformations effectively?
Students can practice graph transformations by working through exercises that involve sketching graphs based on given transformations. Utilizing graphing utilities can also enhance understanding, as students can visualize the effects of transformations in real-time. Additionally, creating tables of values for functions before and after transformations can help solidify the concepts by showing how specific points change. Engaging with a variety of examples will build confidence in applying these concepts.

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