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### Splash Screen Lesson Menu 5 Minute Check 1 5 Minute Check 2 5 Minute Check 3 5 Minute Check 4 5 Minute Check 5 Then/Now Vocabulary Example 1 Example 1 ... – PowerPoint PPT presentation

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Title: Splash Screen

1
Splash Screen
2
Five-Minute Check Then/Now New Vocabulary Example
1 Real-World Example Estimate Function
Values Example 2 Find Domain and Range Example
3 Find y-Intercepts Example 4 Find Zeros Key
Concept Tests for Symmetry Example 5 Test for
Symmetry Key Concept Even and Odd
Functions Example 6 Identify Even and Odd
Functions
3
5Minute 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.
4
5Minute 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.
5
5Minute Check 3
Find the function value for f (2) if f (x) 6
x 2.
A. 2 B. 4 C. 8 D. 10
6
5Minute Check 4
7
5Minute Check 5
8
Then/Now
You identified functions. (Lesson 1-1)
• Use graphs of functions to estimate function
values and find domains, ranges, y-intercepts,
and zeros of functions.
• Explore symmetries of graphs, and identify even
and odd functions.

9
Vocabulary
• zeros
• roots
• line symmetry
• point symmetry
• even function
• odd function

10
Example 1
Estimate Function Values
A. ADVERTISING The function f (x) 5x 2 50x
approximates the profit at a toy company, where
x is marketing costs and f (x) is 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.
11
Example 1
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.
12
Example 1
Estimate Function Values
B. ADVERTISING The function f (x) 5x 2 50x
approximates the profit at a toy company, where
x is marketing costs and f (x) is 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.
13
Example 1
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.
14
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 p (x) 0.325x3 1.5x2 22,
where x is the month, and 0 lt 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
15
Example 2
Find Domain and Range
Use the graph of f to find the domain and range
of the function.
16
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.

17
Example 2
Find Domain and Range
18
Example 2
Use the graph of f to find the domain and range
of the function.
19
Example 3
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.
20
Example 3
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.
21
Example 3
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.
22
Example 3
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.
23
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
24
Example 4
Find Zeros
Use the graph of f (x) x 3 x to approximate
its zero(s). Then find its zero(s) algebraically.
25
Example 4
Find Zeros
Estimate Graphically The x-intercepts appear to
be at about -1, 0, and 1. Solve
Algebraically x 3 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 Solve for x. The zeros of f are 0, 1, and
-1.
26
Example 4
A. 2.5 B. 1 C. 5 D. 9
27
Key Concept 1
28
Example 5
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.
29
Example 5
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.
30
Example 5
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
31
Example 5
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.
32
Example 5
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.
33
Example 5
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
34
Example 5
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
• symmetric with respect to the origin
• D. not symmetric with respect to the x-axis,
y-axis, or the origin

35
Key Concept 2
36
Example 6
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.
37
Example 6
Identify Even and Odd Functions
It appears that the graph of the function is
neither symmetric with respect to the y-axis or
to the origin. Test this conjecture. f (-x)
(-x) 2 4(-x) 4 Substitute -x for x. x 2
4x 4 Simplify.
Since f (x) -x 2 4x - 4, the function is
neither even nor odd because f (-x) ? f (x) or f
(x).
38
Example 6
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.
39
Example 6
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. x 2 - 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
40
Example 6
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.
41
Example 6
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).
42
Example 6
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 or odd
43
End of the Lesson