5Minute Check 5 State the domain of. A. B.(3, 3) C. D.

State the domain of .

A.

B. (–3, 3)

C.

D.

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.

• zeros

• roots

• line symmetry

• point symmetry

• even function

• odd function

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.

Answer:


$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


$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.

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. $25B. $23C. $22D. $20

Find Domain and Range

Use the graph of f to find the domain and range of the function.

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 . In set-builder notation, the domain is .

RangeThe 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 .

Use the graph of f to find the domain and range of the function.

A. Domain:

Range:

B. Domain:

Range:

C. Domain:

Range:

D. Domain:

Range:

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.

Find y-Intercepts

Answer:

Estimate GraphicallyIt appears that f (x) intersects the y-axis at approximately (0, 4), so the y-intercept is about 4.Solve AlgebraicallyFind f (0).f (0) = (0)2 – 4(0) + 4 = 4.The y-intercept is 4.

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.

Find y-Intercepts

Estimate Graphicallyg (x) intersects the y-axis at approximately (0,1), so the y-intercept is about 1.Solve AlgebraicallyFind g (0).g (0) = |0 + 2| – 3 or –1The y-intercept is –1.

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

Find Zeros

Use the graph of f (x) = x 3 – x to approximate its

zero(s). Then find its zero(s) algebraically.

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 Propertyx = 0 x = 1 x = 1 Solve for x.The zeros of f are 0, 1, and 1.

A. –2.5

B. –1

C. 5

D. 9

Use the graph of to approximate its zero(s). Then find its zero(s) algebraically.

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.

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.

Test for Symmetry

Confirm Algebraically Because x2 + 2 is equivalent to (x)2 + 2, the graph is symmetric with respect to the y-axis.

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.

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.

Test for Symmetry

Confirm Algebraically Because (x)(y) = 6 is equivalent to (x)(y) = 6, the graph is symmetric with respect to the origin.

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 originD. not symmetric with respect to the

x-axis, y-axis, or the origin

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.

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 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).

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.



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).

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.



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).

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

• zeros

• roots

• line symmetry

• point symmetry

• even function

• odd function

5Minute Check 5 State the domain of. A. B.(3, 3) C. D.

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