1.2ANALYZING_GRAPHS_OF_FU
NCTIONS_AND_RELATIONS
OBJECTIVES :
• 1.2.1: Use graphs of functions to estimate function values and find domains,
ranges, y-intercepts, and zeros of functions.
• 1.2.2: Explore symmetries of graphs, and identify
even and odd functions.
5–MINUTE CHECK 1
Determine whether 2y + 5x = 7 represents y as a
function of x.
A. y is a function of x.
B. y is not a function of x.
1 minute
5–MINUTE CHECK 2
Determine whether the
graph represents y as a
function of x.
A. y is a function of x.
B. y is not a function of x.
1 minute
5–MINUTE CHECK 3
Find f(–2) if f(x) = 6 – x 2.
A. 2
B. 4
C. 8
D. 10
1 minute
5–MINUTE CHECK 4
Find
A. 3a – 2
B.
C.
1 minute
D.
5–MINUTE CHECK 5
State the domain of
.
A.
B. (–3, 3)
C.
D.
1 minute
EXAMPLE 1A
Estimate Function Values
A. ADVERTISING The function f(x) = –5x 2 + 50x
approximates the profit at a toy company, where
x represents marketing costs and f(x) represents
profit. Both costs and profits are measured in tens
of thousands of dollars. Use the graph to estimate
the profit when marketing costs are $30,000.
Confirm your estimate
algebraically.
EXAMPLE 1A
Estimate Function Values
$30,000 is three ten thousands. The function value at
x = 3 appears to be about 100 ten thousands, so the
total profit was about $1,000,000. To confirm this
estimate algebraically, find f(3).
f(3) = -5(3)2 + 50(3) = 105, or about $1,050,000.
The graphical estimate of about $1,000,000 is
reasonable.
Answer: $1,050,000
EXAMPLE 1B
Estimate Function Values
B. ADVERTISING The function f(x) = –5x 2 + 50x
approximates the profit at a toy company, where
x represents marketing costs and f(x) represents
profit. Both costs and profits are measured in tens
of thousands of dollars. Use the graph to estimate
marketing costs when the profit is $1,250,000.
Confirm your estimate
algebraically.
EXAMPLE 1B
Estimate Function Values
$1,250,000 is 125 ten thousands. The value of the
function appears to reach 125 ten thousands for an
x-value of about 5. This represents an estimate of
5 ● $10,000 or $50,000. To confirm algebraically, find
f(5). f(5) = -5(5)2 + 50(5) = 125, or about $1,250,000.
The graphical estimate that marketing costs are $50,000
when the profit is $1,250,000 is reasonable.
Answer: $50,000
ACTIVITY
• Kindly join this link
https://quizizz.com/join/quiz/5d9744d42cd93d001b266397/start?studentShare=true
5 minutes
EXAMPLE 1
PROFIT A-Z Toy Boat Company found the average price
of its boats over a six month period. The average price
for each boat can be represented by the polynomial
function p (x) = –0.325x3 + 1.5x2 + 22, where x is the
month, and 0 < x ≤ 6.
Use the graph to estimate the average price of a boat in
the fourth month. Confirm you estimate algebraically.
A. $25
B. $23
C. $22
D. $20
EXAMPLE 2
Find Domain and Range
Use the graph of f to find the domain and range of
the function.
EXAMPLE 2
Find Domain and Range
Domain
•
The dot at (3, -3) indicates that the domain of
f ends at 3 and includes 3.
•
The arrow on the left side indicates that the graph
will continue without bound.
The domain of f is
domain is
. In set-builder notation, the
.
Range
The graph does not extend above y = 2, but f(x)
decreases without bound for smaller and smaller
values of x. So the range of f is
.
EXAMPLE 2
Find Domain and Range
Answer: D:
R:
EXAMPLE 2
Use the graph of f to find the domain and range of
the function.
A. Domain:
Range:
B. Domain:
Range:
C. Domain:
Range:
1 minute
D. Domain:
Range:
ACTIVITY
• https://create.kahoot.it/share/domain-and-range-of-functions/683c3cac-d064-42a0-b631a8abcb16e70a
5 minutes
EXAMPLE 3A
Find y-Intercepts
A. Use the graph of the function f(x) = x 2 – 4x + 4
to approximate its y-intercept. Then find the
y-intercept algebraically.
EXAMPLE 3A
Find y-Intercepts
Estimate Graphically
It appears that f(x) intersects the y-axis at
approximately (0, 4), so the y-intercept is about 4.
Solve Algebraically
Find f(0).
f(0) = (0)2 – 4(0) + 4 = 4.
The y-intercept is 4.
Answer: 4
EXAMPLE 3B
Find y-Intercepts
B. Use the graph of the function g(x) =│x + 2│– 3
to approximate its y-intercept. Then find the
y-intercept algebraically.
EXAMPLE 3B
Find y-Intercepts
Estimate Graphically
g(x) intersects the y-axis at approximately (0, -1), so
the y-intercept is about -1.
Solve Algebraically
Find g(0).
g(0) = |0 + 2| – 3 or –1
The y-intercept is –1.
Answer: -1
EXAMPLE 3
Use the graph of the function to approximate its
y-intercept. Then find the y-intercept algebraically.
A. –1; f(0) = –1
B. 0; f(0) = 0
C. 1; f(0) = 1
D. 2; f(0) = 2
1 minute
EXAMPLE 4
Find Zeros
Use the graph of f(x) = x 3 – x to approximate its
zero(s). Then find its zero(s) algebraically.
3 minutes
EXAMPLE 4
Find Zeros
Estimate Graphically
The x-intercepts appear to be at about -1, 0, and 1.
Solve Algebraically
x3 – x = 0
Let f(x) = 0.
x(x 2 – 1) = 0
Factor.
x(x – 1)(x + 1) = 0
Factor.
x = 0 or x – 1 = 0 or x + 1 = 0
Zero Product Property
x=0
x =1
x = -1
The zeros of f are 0, 1, and -1.
Answer: -1, 0, 1
Solve for x.
EXAMPLE 4
Use the graph of
to approximate
its zero(s). Then find its zero(s) algebraically.
2 minutes
A. –2.5
B. –1
C. 5
D. 9
KEY CONCEPT 1
EXAMPLE 5A
Test for Symmetry
A. Use the graph of the equation y = x 2 + 2 to test
for symmetry with respect to the x-axis, the y-axis,
and the origin. Support the answer numerically.
Then confirm algebraically.
EXAMPLE 5A
Test for Symmetry
Analyze Graphically
The graph appears to be symmetric with respect to the
y-axis because for every point (x, y) on the graph,
there is a point (-x, y).
Support Numerically
A table of values supports this conjecture.
EXAMPLE 5A
Test for Symmetry
Confirm Algebraically
Because x2 + 2 is equivalent to (-x)2 + 2, the graph is
symmetric with respect to the y-axis.
Answer: symmetric with respect to the y-axis
EXAMPLE 5B
Test for Symmetry
B. Use the graph of the equation xy = –6 to test for
symmetry with respect to the x-axis, the y-axis,
and the origin. Support the answer numerically.
Then confirm algebraically.
EXAMPLE 5B
Test for Symmetry
Analyze Graphically
The graph appears to be symmetric with respect to the
origin because for every point (x, y) on the graph,
there is a point (-x, -y).
Support Numerically
A table of values supports this conjecture.
EXAMPLE 5B
Test for Symmetry
Confirm Algebraically
Because (-x)( -y) = -6 is equivalent to (x)(y) = -6, the
graph is symmetric with respect to the origin.
Answer: symmetric with respect to the origin
EXAMPLE 5
2 minutes
Use the graph of the equation
y = –x 3 to test for symmetry
with respect to the x-axis, the
y-axis, and the origin. Support
the answer numerically. Then
confirm algebraically.
A.
symmetric with respect to the x-axis
B.
symmetric with respect to the y-axis
C.
symmetric with respect to the origin
D.
not symmetric with respect to the
x-axis, y-axis, or the origin
KEY CONCEPT 2
EXAMPLE 6A
Identify Even and Odd Functions
A. Graph the function f(x) = x 2 – 4x + 4 using a
graphing calculator. Analyze the graph to
determine whether the function is even, odd, or
neither. Confirm algebraically. If even or odd,
describe the symmetry of the graph of the
function.
EXAMPLE 6A
Identify Even and Odd Functions
It appears that the graph of the function is neither
symmetric with respect to the y-axis nor to the origin.
Test this conjecture algebraically.
f(-x) = (-x) 2 – 4(-x) + 4
= x 2 + 4x + 4
Substitute -x for x.
Simplify.
Since –f(x) = -x 2 + 4x - 4, the function is neither even
nor odd because f(-x) ≠ f(x) or –f(x).
Answer: neither
EXAMPLE 6B
Identify Even and Odd Functions
B. Graph the function f(x) = x 2 – 4 using a
graphing calculator. Analyze the graph to
determine whether the function is even, odd, or
neither. Confirm algebraically. If even or odd,
describe the symmetry of the graph of the
function.
EXAMPLE 6B
Identify Even and Odd Functions
From the graph, it appears that the function is
symmetric with respect to the y-axis. Test this
conjecture algebraically.
f(-x) = (-x)2 – 4
Substitute -x for x.
= x2 - 4
Simplify.
= f(x)
Original function f(x) = x 2 – 4
The function is even because f(-x) = f(x).
Answer: even; symmetric with respect to the y-axis
EXAMPLE 6C
Identify Even and Odd Functions
C. Graph the function f(x) = x 3 – 3x 2 – x + 3 using a
graphing calculator. Analyze the graph to
determine whether the function is even, odd, or
neither. Confirm algebraically. If even or odd,
describe the symmetry of the graph of the
function.
EXAMPLE 6C
Identify Even and Odd Functions
From the graph, it appears that the function is neither
symmetric with respect to the y-axis nor to the origin.
Test this conjecture algebraically.
f(–x) = (–x) 3 – 3(–x)2 – (–x) + 3 Substitute –x for x.
= –x 3 – 3x 2 + x + 3
Simplify.
Because –f(x) = –x 3 + 3x 2 + x – 3, the function is
neither even nor odd because f(–x) ≠ f(x) or –f(x).
Answer: neither
EXAMPLE 6
2 minutes
Graph the function f(x) = x 4 – 8 using a graphing
calculator. Analyze the graph to determine whether
the graph is even, odd, or neither. Confirm
algebraically. If even or odd, describe the
symmetry of the graph of the function.
A. odd; symmetric with
respect to the origin
B. even; symmetric with
respect to the y-axis
C. neither even nor odd
ACTIVITY
•https://create.kahoot.it/share/even-odd-functions/93c68eff9a29-479e-a3a0-3a7f977588e6
10 minutes