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Analysis of Graphs of
Functions
A FIGURE HAS rotational symmetry around an axis I if it coincides with
itself by all rotations about I. Because of their complete rotational symmetry,
the circle in the plane and the sphere in space were considered by the early
Greeks to be the most perfect geometric figures. Aristotle assumed a
spherical shape for the celestial bodies because any other would detract from
their heavenly perfection.
Symmetry has been an important characteristic of art from the earliest
times. The art of M.C. Escher (1898–1972) is composed of symmetries and
translations. Leonardo da Vinci’s sketches indicate a superior understanding
of symmetry. Almost all nature exhibits symmetry—from the hexagons of
snowflakes to the diatom, a microscopic sea plant. Perhaps the most striking
examples of symmetry in nature are crystals. The photos on this page show a
diatom and a cross section of a tourmaline.
This chapter introduces, among other things, how symmetry is found in
the graphs of certain functions.
CHAPTER OUTLINE
2.1 Graphs of Basic
Functions and Relations;
Symmetry
2.2 Vertical and Horizontal
Shifts of Graphs
2.3 Stretching, Shrinking,
and Ref lecting Graphs
2.4 Absolute Value
Functions: Graphs,
Equations, Inequalities,
and Applications
2.5 Piecewise-Defined
Functions
2.6 Operations and
Composition
(Source: Mathematics, Life Science Library,
Time Inc., New York, 1963.)
1
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GRAPHS OF BASIC FUNCTIONS AND RELATIONS; SYMMETRY
Continuity ● Increasing and Decreasing Functions ● The Identity Function ● The Squaring Function and Symmetry with
Respect to the y-Axis ● The Cubing Function and Symmetry with Respect to the Origin ● The Square Root and Cube Root
Functions ● The Absolute Value Function ● The Relation x y2 and Symmetry with Respect to the x-Axis ● Even and
Odd Functions
Continuity
In Chapter 1, our work dealt mainly with linear functions. The graph of a linear function, a straight line, may be drawn by hand over any interval of its domain without
picking the pencil up from the paper. In mathematics we say that a function with
this property is continuous over any interval. The formal definition of continuity requires concepts from calculus, but we can give an informal definition at the precalculus level.
Continuity
A function is continuous over an interval of its domain if its hand-drawn
graph over that interval can be sketched without lifting the pencil from
the paper.
If a function is not continuous at a point, then it may have a point of discontinuity (Figure 1(a)), or it may have a vertical asymptote (a vertical line that the graph does
not intersect, as in Figure 1(b)). Asymptotes will be discussed further in Chapter 4.
y
y
y
x
0
FIGURE 2
0
x
0
2
x
2
y
Both functions are
discontinuous at x = 2.
0
(a)
x
3
(b)
FIGURE 1
FIGURE 3
E X A M P L E 1 Determining Intervals of Continuity
Figures 2, 3, and 4 show graphs of functions. Indicate the intervals of the domain over
which they are continuous.
y
2
0
–2
FIGURE 4
x
Solution The function in Figure 2 is continuous over the entire domain of real
numbers, , . The function in Figure 3 has a point of discontinuity at x 3. It is
continuous over the interval , 3 and the interval 3, . Finally, the function in
Figure 4 has a vertical asymptote at x 2, as indicated by the dashed line. It is continuous over the intervals , 2 and 2, .
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3
Graphs of Basic Functions and Relations; Symmetry
Increasing and Decreasing Functions
If a continuous function is not constant over an interval, then its graph will either rise
from left to right or fall from left to right or some combination of both. We use the
words increasing and decreasing to describe this behavior. For example, a linear function with positive slope is increasing over its entire domain, while one with negative
slope is decreasing. See Figure 5.
y
y
f (x) = mx + b, m > 0
f (x) = mx + b, m < 0
(a)
x
0
x
0
f is decreasing, since it falls
from left to right.
f is increasing, since it rises
from left to right.
(b)
FIGURE 5
FIGURE 6
The table feature of a graphing calculator can be used to illustrate the concepts of
increasing and decreasing functions. As seen in Figure 6(a), the graph of Y1 x 1
has positive slope m 1, and as x-values increase, range values increase. The graph
of Y2 x 1 has negative slope m 1, and as x-values increase, range values
decrease, as shown in Figure 6(b).
Informally speaking, a function increases on an interval of its domain if its graph
rises from left to right. A function decreases on an interval of its domain if its
graph falls from left to right. A function is constant on an interval of its domain if its
graph is horizontal on the interval. (While some functions are increasing, decreasing,
or constant over their entire domains, many are not. Hence, we speak of increasing,
decreasing, or constant over an interval of the domain.)
Looking Ahead to Calculus
In calculus, determining where a
function increases, decreases, or is
constant is important in curvesketching. The derivative of a function
provides a formula for determining the
slope of a line tangent to the curve. If
the slope is positive for a given
domain value, the function is
increasing at that point; if the slope is
negative, the function is decreasing;
and if the slope is 0, the function is
constant. For example, look at
Figure 7(a). Choose a point on the
curve between x1, f x1 and
x2, f x2, and draw a line tangent to
the curve at that point. The line rises
from left to right, so its slope is
positive. This positive slope indicates
that the function is increasing at that
point. (Extend this idea to Figures 7(b)
and 7(c).)
Criteria for Increasing, Decreasing, and Constant
Suppose that a function f is defined over an interval I.
(a) f increases on I if, whenever x1 x2 , fx1 fx2;
(b) f decreases on I if, whenever x1 x2 , fx1 fx2;
(c) f is constant on I if, for every x1 and x2 , fx1 fx2.
Figure 7 illustrates these ideas.
y
y
f (x 2 )
y
f (x1) = f (x 2 )
f (x1)
f (x1)
x1
0
x2
x
f (x 2 )
0
x1
x2
x
x1
0
x2
x
When x 1 < x 2, f (x 1 ) < f (x 2 ).
f is increasing.
When x 1 < x 2, f (x 1 ) > f (x 2 ).
f is decreasing.
For x 1 and x 2, f (x 1 ) = f (x 2 ).
f is constant.
(a)
(b)
(c)
FIGURE 7
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N O T E To decide whether a function is increasing, decreasing, or constant on an interval,
ask yourself “What does y do as x goes from left to right?”
E X A M P L E 2 Determining Intervals over Which a Function Is Increasing,
Decreasing, or Constant
Figure 8 shows the graph of a function. Determine the intervals over which the function is increasing, decreasing, or constant.
y
6
2
–2
0
x
2
FIGURE 8
FOR DISCUSSION
1.
2.
3.
In the standard viewing
window of your calculator,
enter any linear function
defined by y mx b
with m 0 . Now, trace the
graph from left to right.
Watch the y-values as the
x-values get larger. What is
happening to y? How does
this reinforce the concepts
presented so far in this
section?
Repeat this exercise, but with
m 0.
Repeat this exercise, but with
m 0.
Solution We should ask, “What is happening to the y-values as x-values are getting larger?” Moving from left to right on the graph, we see that on the interval
, 1, the y-values are decreasing; on the interval 1, 3, the y-values are increasing;
and on the interval 3, , the y-values are constant (and equal to 6). Therefore, the function is decreasing on , 1, increasing on 1, 3, and constant on 3, .
C A U T I O N A common error involves writing range values ( y-values) when determining intervals like those in Example 2. Remember that we are determining intervals of the domain, and
thus are interested in x-values for our interval designations. Range values are not written when
designating such intervals.
The next part of this section introduces several important basic functions and
relations.
The Identity Function
If we let m 1 and b 0 in the general form of the linear function defined by
fx mx b, we get the identity function, or fx x. This function pairs every
real number with itself. See Figure 9.
FUNCTION CAPSULE
IDENTITY FUNCTION
Domain: , f(x) x
Range: , y
f (x) = x
10
f (x) = x
2
0
x
2
–10
10
–10
FIGURE 9
●
●
fx x is increasing on its entire domain, , .
It is continuous on its entire domain, , .
The Squaring Function and Symmetry with Respect to the y-Axis
The degree of a polynomial function in the variable x is the greatest exponent on x in
the polynomial. The identity function is a degree 1 function. The squaring function,
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Looking Ahead to Calculus
Calculus allows us to find areas of
regions in the plane. For example, we
can find the area of the region below
the graph of y x 2 , above the
x-axis, bounded on the left by the line
x 2 , and bounded on the right by
x 2 . Draw a sketch of this region,
and notice that due to the symmetry
of the graph of y x 2 , the desired
area is twice that of the area to the
right of the y-axis. Thus, symmetry
can be used to reduce the original
problem to an easier one by simply
finding the area to the right of the
y-axis, and then doubling the answer.
Graphs of Basic Functions and Relations; Symmetry
5
defined by fx x 2, is the simplest degree 2 (or quadratic) function. (The word
quadratic refers to degree 2.) See Figure 10.
The squaring function pairs every real number with its square. Its graph is called
a parabola. The point at which the graph changes from decreasing to increasing (the
point 0, 0) is called the vertex of the graph. (For a parabola that opens downward,
the vertex is the point at which the graph changes from increasing to decreasing.)
FUNCTION CAPSULE
SQUARING FUNCTION
Domain: , f(x) x2
Range: 0, y
–2 –1
f (x) = x 2
f (x) = x 2
4
3
2
1
0
x
1 2
10
–10
10
–10
FIGURE 10
●
●
fx x 2 decreases on the interval , 0 and increases on the interval 0, .
It is continuous on its entire domain, , .
If we were able to “fold” the graph of fx x 2 along the y-axis, the two halves
would coincide exactly. In mathematics we refer to this property as symmetry, and we
say that the graph of fx x 2 is symmetric with respect to the y-axis.
Symmetry with Respect to the y-Axis
If a function f is defined so that
f(x) f(x)
for all x in its domain, then the graph of f is symmetric with respect to
the y-axis.
To illustrate that the graph of fx x 2 is symmetric with respect to the y-axis,
note that
f4 f4 16
f2 f2 4
f3 f3 9
f1 f1 1.
This pattern holds for any real number x, since
This table supports, but does
not prove, that the graph of the
squaring function is symmetric
with respect to the y-axis.
fx x2 x 2 fx.
In general, if a graph is symmetric with respect to the y-axis, the following is true: If
a, b is on the graph, so is a, b.
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The Cubing Function and Symmetry with Respect to the Origin
The cubing function, defined by fx x 3, is the simplest degree 3 function. It pairs
with each real number the third power, or cube, of the number. See Figure 11. The
point at which the graph of the cubing function changes from “opening downward” to
“opening upward” (the point 0, 0) is called an inflection point.
FUNCTION CAPSULE
CUBING FUNCTION
Domain: , f(x) x3
Range: , y
f (x) = x3
f (x) = x3
8
10
0
x
2
–10
10
–10
FIGURE 11
●
●
fx x 3 increases on its entire domain, , .
It is continuous on its entire domain, , .
If we were able to “fold” the graph of fx x 3 along the y-axis and then along
the x-axis, forming a “corner” at the origin, the two parts of the graph would coincide
exactly. We say that the graph of fx x 3 is symmetric with respect to the origin.
Symmetry with Respect to the Origin
If a function f is defined so that
f(x) f(x)
for all x in its domain, then the graph of f is symmetric with respect to
the origin.
To illustrate that the graph of fx x 3 is symmetric with respect to the origin,
note that
f2 f2 8
f1 f1 1.
This pattern holds for any real number x, since
This table supports, but does
not prove, that the graph of the
cubing function is symmetric
with respect to the origin.
fx x3 13x 3 x 3 fx.
In general, if a graph is symmetric with respect to the origin, the following is true: If
a, b is on the graph, so is a, b.
E X A M P L E 3 Determining Symmetry
Show that (a) fx x 4 3x 2 8 has a graph that is symmetric with respect to the
y-axis; (b) fx x 3 4x has a graph that is symmetric with respect to the origin.
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Analytic Solution
(a) We must show that fx fx for any x.
Graphs of Basic Functions and Relations; Symmetry
Graphing Calculator Solution
(a) The graph in Figure 12 supports the analytic proof.
fx x 3x 8
4
2
f (x) = x 4 – 3x 2 – 8
Substitute x for x.
10
1 x 31 x 8
x 4 3x 2 8
fx
4 4
7
2 2
–10
This proves that the graph is symmetric with respect to the y-axis.
(b) We must show that fx fx for any x.
10
–10
This graph is symmetric with respect to the y-axis.
FIGURE 12
fx x3 4x
(b) The graph in Figure 13 supports the analytic proof.
Substitute x for x.
f (x) = x3 – 4x
13x 3 4x
x 3 4x
*
3
x 4x
fx
10
–10
In the line denoted *, note that the signs of the
coefficients are all opposites of those in fx. We
completed the solution by factoring out 1,
showing that the final result is fx.
10
–10
This graph is symmetric with respect to the origin.
FIGURE 13
The Square Root and Cube Root Functions
Some functions involve expressions with radicals. One of these is the square root
function, defined by fx x. See Figure 14. Notice that for the function value to
be a real number, we must have x 0. Thus, the domain is restricted to 0, .
FUNCTION CAPSULE
f(x) x
SQUARE ROOT FUNCTION
Domain: 0, Range: 0, y
f (x) = √x
10
f (x) = √x
2
1
0
x
–10
10
1 2 3 4
–10
FIGURE 14
●
●
fx x increases on its entire domain, 0, .
It is continuous on its entire domain, 0, .
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T ECH N OLO GY N O T E
The definition of a rational exponent
allows us to enter x as x 1/2 and
3
x as x 1/3 when using a calculator.
Further discussion of rational
exponents can be found in Chapter R.
3
x (Figure 15), differs from the
The cube root function, defined by fx square root function in that any real number—positive, 0, or negative— has a real
cube root, and thus the domain is , .
When
when
when
3
x 0,
3
x 0,
3
x 0.
x 0,
x 0,
x 0,
and
As a result, the range is also , .
FUNCTION CAPSULE
CUBE ROOT FUNCTION
Domain: , 3
f(x) x
Range: , y
3
f (x) = √x
10
3
f (x) = √x
–8
0
2
x
–10
10
8
–10
FIGURE 15
●
●
T ECH N OLO GY N O T E
You should become familiar with the
command on your particular calculator
that allows you to graph the absolute
value function. Consult the graphing
calculator manual that accompanies
this text or your owner’s manual.
3
fx x increases on its entire domain, , .
It is continuous on its entire domain, , .
The Absolute Value Function
On a number line, the absolute value of a real number x, denoted x, represents its
undirected distance from the origin, 0. The absolute value function, which pairs
every real number with its absolute value, is defined as follows.
Absolute Value x
f(x) x x
x
if x > 0
if x < 0
Notice that this function is defined in two parts. We use x x if x is positive or
0, and we use x x if x is negative. Since x can be any real number, the domain of
the absolute value function is , , but since x cannot be negative, the range is
0, . See Figure 16.
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Graphs of Basic Functions and Relations; Symmetry
9
FUNCTION CAPSULE
f(x) x
ABSOLUTE VALUE FUNCTION
Domain: , Range: 0, y
f (x) = | x|
10
f (x) = | x||
2
0
–10
x
10
1 2 3 4
–10
FIGURE 16
●
●
fx x decreases on the interval , 0 and increases on 0, .
It is continuous on its entire domain, , .
FOR DISCUSSION
Based on the functions discussed so far in this section, answer each question.
1.
2.
3.
4.
Which functions have graphs that are symmetric with respect to the y-axis?
Which functions have graphs that are symmetric with respect to the origin?
Which functions have graphs with neither of these symmetries?
Why is it not possible for the graph of a nonzero function to be symmetric with
respect to the x-axis?
The Relation x y2 and Symmetry with Respect to the x-Axis
Recall that a function is a relation where every domain value is paired with one and
only one range value. However, there are cases where we are interested in graphing relations that are not functions, and one of the simplest of these is the relation defined
by the equation x y 2. Notice that the table of selected ordered pairs below indicates
that this relation has two different y-values for each positive value of x. If we plot these
points and join them with a smooth curve, we find that the graph of x y 2 is a
parabola opening to the right. See Figure 17.
Selected Ordered Pairs for x y2
x
y
0
1
4
9
0
1
2
3
y
x = y2
3
Two different y-values
for the same x-value
–2
–4
0
x
4
9
FIGURE 17
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T ECH N OLO GY N O T E
You should be aware of how function
mode (as opposed to parametric
mode) is activated for your model.
If a graphing calculator is set in function mode, it is not possible to graph
x y 2 directly. (However, if it is set in parametric mode, such a curve is possible with
direct graphing.) To overcome this problem, we begin with x y 2 and take the square
root on each side, remembering to choose both the positive and negative square roots
of x.
x y2
y2 x
y x
Given equation
Transform so that y is on the left.
Take square roots.
Now, we have x y defined by two functions, Y1 x and Y2 x. Entering
both of these into a calculator gives the graph shown in Figure 18(a).
2
Y1 = √x
Y2 = –√x
10
–10
10
–10
The graph of the relation x = y 2
is symmetric with respect to the
x-axis.
(a)
This screen shows selected ordered
pairs for the relation x = y 2, where
Y1 = √x and Y2 = – √x.
(b)
FIGURE 18
If we were to “fold” the graph of x y 2 along the x-axis, the two halves of the
parabola would coincide. This graph exhibits symmetry with respect to the x-axis. A
table supports this symmetry, as seen in Figure 18(b).
Symmetry with Respect to the x-Axis
If replacing y with y in an equation results in the same equation, then
the graph is symmetric with respect to the x-axis.
To illustrate this, begin with x y 2 and replace y with y.
x y2
x 12y 2
x y2
The result is the same equation.
In general, if a graph is symmetric with respect to the x-axis, the following is true: If
a, b is on the graph, so is a, b.
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Graphs of Basic Functions and Relations; Symmetry
11
A summary of the types of symmetry just discussed follows.
Type of Symmetry
Basic Fact about Points
on the Graph
Example
If a, b is on the graph, so
is a, b.
y
y-axis symmetry
(a, b)
(–a, b)
x
0
Origin symmetry
FOR DISCUSSION
The three functions discussed
in Example 4 on the next page are
graphed in Figure 19, but not necessarily in the same order as in the
example. Without actually using
your calculator, identify each
function.
If a, b is on the graph, so
is a, b.
y
(a, b)
0
x
(–a, – b)
x-axis symmetry
(not possible
for a nonzero
function)
10
If a, b is on the graph, so
is a, b.
y
(a, b)
x
0
(a, – b)
–4
4
Even and Odd Functions
–10
The concepts of symmetry with respect to the y-axis and symmetry with respect to the
origin are closely associated with the concepts of even and odd functions.
(a)
3
Even and Odd Functions
–4
4
A function f is called an even function if fx fx for all x in the
domain of f. (Its graph is symmetric with respect to the y-axis.)
A function f is called an odd function if fx fx for all x in the
domain of f. (Its graph is symmetric with respect to the origin.)
–1
(b)
15
As an illustration, observe the following.
fx x 2 is an even function because
–4
4
–5
(c)
FIGURE 19
fx x2
12x 2
x2
fx.
fx x 3 is an odd function because
fx x3
13x 3
x 3
fx.
A function may be neither even nor odd; for example, fx x is neither even
nor odd.
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E X A M P L E 4 Determining Whether Functions are Even, Odd, or Neither
Decide if each function defined is even, odd, or neither.
(a) fx 8x 4 3x 2
(b) fx 6x 3 9x
(c) fx 3x 2 5x
Solution
(a) Replacing x with x gives
fx 8x4 3x2 8x 4 3x 2 fx.
Since fx fx for each x in the domain of the function, f is even.
(b) Here
fx 6x3 9x 6x 3 9x fx.
The function is odd.
(c) fx 3x2 5x
3x 2 5x
Since fx fx and fx fx, f is neither even nor odd.
2.1
EXERCISES
Concept Check
Fill in each blank with the correct response.
1. The domain and the range of the identity function are both
.
2. The domain of the squaring function is
while its range is
.
,
3. The graph of the cubing function changes from “opening
downward” to “opening upward” at the point
.
4. The domain of the square root function is
and its range is
.
5. The cube root function
tire domain.
(increases/decreases)
,
on its en-
6. The absolute value function decreases on the interval
and increases on the interval
.
7. The graph of the relation given by x y 2 is symmetric
with respect to the
.
8. The function defined by f x x 4 x 2 is an
(even/odd)
function.
9. The function defined by f x x 3 x is an
(even/odd)
function.
10. If a function is even, its graph is symmetric with respect to
the
; if it is odd, its graph is symmetric with
respect to the
.
Determine the intervals of the domain over which the given function is continuous.
11.
y
y
12.
y
13.
(0, 3)
0
14.
0
x
y
15.
0
x
16.
y
x
y
(1, 2)
0
0
x
(0, –1)
–3
x
0
x
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13
Graphs of Basic Functions and Relations; Symmetry
Determine the intervals of the domain over which the given function is (a) increasing, (b) decreasing, and (c) constant. Then give the
(d) domain and (e) range.
y
17.
y
19.
(– 1, 3)
x
0
(1, 3)
(4, 3)
x
(3, 0)
21.
(4, 3)
x
0
0
y
20.
y
18.
y
22.
y
(3, 3)
(3, 1.5)
(– 3, 3)
(– 2, 2)
0
x
(3, 1)
x
0
(– 2, – 2)
x
0
(3, – 2)
In Exercises 23–28, graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then, fill in the blank of this sentence with either increasing or decreasing:
OVER THE INTERVAL SPECIFIED, THIS FUNCTION IS
.
23. f x x ; , 24. f x x ; , 25. f x x 4; , 0
26. f x x 4; 0, 27. f x x; , 0
28. f x x; 0, 5
3
Using visual observation, determine whether each graph is symmetric with respect to the (a) x-axis, (b) y-axis, (c) origin.
y
29.
y
30.
0
0
x
0
35.
y
33.
y
x
– 2␲
36.
2␲
–4
34.
x
0
4
x
x
0
32.
y
31.
0
4
– 2␲
2␲
–4
y
x
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Analysis of Graphs of Functions
Concept Check In Exercises 37– 42, you are given a calculator-generated graph of a relation that exhibits a type of symmetry.
Based on the given information at the bottom of each figure, determine the coordinates of another point that must also lie on the graph.
37. symmetric with respect
to the y-axis
38. symmetric with respect
to the y-axis
8
10
– 10
3.1
– 12
10
12
– 4.7
4.7
– 3.1
–8
– 10
40. symmetric with respect
to the x-axis
41. symmetric with respect
to the origin
42. symmetric with respect
to the origin
4
3.1
–4.7
10
– 2␲
4.7
2␲
– 10
10
– 10
–4
– 3.1
43. Concept Check
function.
39. symmetric with respect
to the x-axis
Complete the table if f is an even
x
3
2
1
1
2
3
g(x)
5
3
2
0
2
3
5
21
25
12
(a) f x f x
Complete the table if g is an odd
x
f(x)
45. Concept Check Complete the left half of the graph of
y f x in the figure for each of the following conditions:
(b) f x f x.
44. Concept Check
function.
13
5
1
46. Concept Check Complete the right half of the graph of
y f x in the figure for each of the following conditions:
y
y
(a) f is odd
2
(b) f is even.
2
0
–2
x
0
2
x
Based on the ordered pairs seen in each pair of tables, make a conjecture about whether the function defined in Y1 is even, odd, or
neither even nor odd.
47.
48.
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2.1
Graphs of Basic Functions and Relations; Symmetry
15
Prove analytically that each function is even.
50. f x 2x 6 8x 2
49. f x x 4 7x 2 6
Prove analytically that each function is odd.
51. f x 3x 3 x
52. f x x 5 2x 3 3x
Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the y-axis,
symmetric with respect to the origin, or neither. Use your calculator to support your conclusion, using the standard window.
53. f x x 3 2x
54. f x x 5 2x 3
55. f x .5x 4 2x 2 1
56. f x .75x 2 x 1
57. f x x 3 x 3
58. f x x 4 5x 2
9 60.
59. Refer to Exercises 53– 58, and determine which exercises
have functions that are:
(a) even
(b) odd
(c) neither even nor odd.
Write an explanation of the connections between symmetry with respect to the y-axis, symmetry with respect to the
origin, even functions, and odd functions.
In Exercises 61 and 62, you are given a portion of the graph of a function near the origin, along with information regarding symmetry of the graph of the complete function. Sketch by hand a simulated calculator graph, using the window given, to show what each
graph would look like.
61. Symmetric with respect to the y-axis;
window: 5, 5 by 0, 5
62. Symmetric with respect to the origin;
window: 5, 5 by 2, 2
2
5
–5
–5
5
5
–2
0
RELATING CONCEPTS
For individual or group investigation (Exercises 63–66)
The line in the sketch and the line on the screen are said to be tangent to the curve at point P.
5
P
–5
5
–2
63. Sketch by hand the graph of y x 2, and choose a point at
which the function is decreasing. Draw a tangent line at that
point. Is the slope of this line positive, negative, or zero?
9 66. Based on your answers to Exercises 63 – 65, what con-
64. Repeat Exercise 63, but choose a point at which the function is increasing.
clusions can be drawn? These conclusions can be verified
using methods discussed in calculus.
65. Repeat Exercise 63, but choose the point at which the
function changes from decreasing to increasing.
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16
9 67.
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Analysis of Graphs of Functions
9 68.
Graph the function defined by
Graph the function defined by
x 9
x3
2
y
y
in the standard window of your calculator. Try to locate
the point for which x 3. What happens? Why do you
think this happens? (Functions of this kind, called rational
functions, will be studied in detail in Chapter 4.)
2.2
x2 9
x3
in the standard window of your calculator. Do you think
that determining continuity strictly by observation of a
calculator-generated graph is foolproof?
VERTICAL AND HORIZONTAL SHIFTS OF GRAPHS
Vertical Shifts ● Horizontal Shifts ● Combinations of Vertical and Horizontal Shifts
Range ● Horizontal Shifts Applied to Equations for Modeling
●
Effects of Shifts on Domain and
In this section we examine how the graphs of the basic functions introduced in the previous section can be shifted vertically and horizontally in the plane.
Vertical Shifts
FOR DISCUSSION
T ECH N OLO GY N O T E
This screen shows how to minimize
keystrokes in the activity in the “For
Discussion” box. The entry for Y1 can
be altered as needed.
In each group, we give four related functions. Graph the four functions in the first
group (Group A), and then answer the questions regarding those functions. Repeat the
process for Group B, Group C, and Group D. Use the standard viewing window in
each case.
A
B
C
D
2
3
3
y1 x
y1 x
y1 x
y1 x
2
3
3
y2 x 3
y2 x 3
y2 x 3
x3
y2 2
3
3
y3 x 2
y3 x 2
y3 x 2
y3 x 2
2
3
3
y4 x 5
y4 x 5
y4 x 5
y4 x5
How does the graph of y2 compare to the graph of y1?
How does the graph of y3 compare to the graph of y1?
How does the graph of y4 compare to the graph of y1?
If c 0, how do you think the graph of y fx c would compare to the graph
of y fx?
5. If c 0, how do you think the graph of y fx c would compare to the graph
of y fx?
1.
2.
3.
4.
Choosing your own value of c, support your answers to Items 4 and 5 graphically.
(Be sure that your choice is appropriate for the standard window.)
The objective of the preceding discussion activity was to make conjectures about
how adding or subtracting a positive constant c affects the graph of a function. In each
case, we obtained a vertical shift, or translation, of the graph of the basic function
with which we started. Although our observations were based on the graphs of four
different basic functions, they can be generalized to any function.
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