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Warm Up

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Evaluate and Graph Polynomial Functions

Warm Up

Lesson Presentation

Lesson Quiz

Warm-Up

Evaluate the expression when x=â€“4.

2.â€“3x3 â€“ 2x2 + 10

1.x2 + 5x

170

â€“4

ANSWER

ANSWER

3. The expressionx2 â€“4represents the amount of matting in square inchesthat is needed to mat a picture. How much matting is needed ifx=6?

32 in.2

ANSWER

1

a. h (x) = x4 â€“ x2 + 3

Example 1

Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.

4

SOLUTION

a. The function is a polynomial function that is

already written in standard form. It has degree 4

(quartic) and a leading coefficient of 1.

b. The function is a polynomial function written as

g(x) = Ï€x2 + 7x â€“ 3 in standard form. It has

degree 2(quadratic) and a leading coefficient of Ï€.

b. g (x) = 7x â€“ 3 + Ï€x2

Example 1

Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.

SOLUTION

Example 1

Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.

c. f (x) = 5x2 + 3x â€“1â€“ x

SOLUTION

c. The function is not a polynomial function

because the term 3x â€“ 1 has an exponent that is not

a whole number.

Example 1

d. k (x) = x + 2xâ€“ 0.6x5

SOLUTION

d. The function is not a polynomial function

because the term 2xdoes not have a variable

base and an exponent that is a whole number.

Example 2

Use direct substitution to evaluatef (x) = 2x4 â€“ 5x3 â€“4x + 8whenx = 3.

f (x) = 2x4 â€“5x3 â€“ 4x+ 8

Write original function.

f (3) = 2(3)4 â€“ 5(3)3 â€“ 4(3) + 8

Substitute 3 for x.

= 162 â€“ 135 â€“ 12 + 8

Evaluate powers and multiply.

= 23

Simplify

Guided Practice

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

1. f (x) = 13 â€“ 2x

ANSWER

polynomial function; f (x) = â€“2x + 13;degree 1, type: linear, leading coefficient: â€“2

Guided Practice

2. p (x) = 9x4 â€“ 5x â€“ 2 + 4

ANSWER

not a polynomial function

3. h (x) = 6x2 + Ï€ â€“ 3x

ANSWER

polynomial function; h(x) = 6x2 â€“ 3x + Ï€;degree 2, type: quadratic, leading coefficient: 6

Guided Practice

Use direct substitution to evaluate the polynomial function for the given value of x.

4. f (x) = x4 + 2x3 + 3x2 â€“ 7; x = â€“2

ANSWER

5

5. g(x) = x3â€“ 5x2 + 6x + 1; x = 4

ANSWER

9

Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

STEP 1

Example 3

Use synthetic substitution to evaluate f (x) from Example2 when x = 3.f (x) = 2x4 â€“ 5x3 â€“4x + 8

SOLUTION

STEP 2

Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add.

Example 3

Synthetic substitution gives f(3) = 23, which matches the result in Example 2.

ANSWER

STEP 3

Multiply the previous sum by the x-value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x-value.

Example 3

ANSWER

The correct answer is D.

Example 4

From the graph, f (x)â†’ â€“âˆž as xâ†’ â€“âˆž and f (x)â†’ â€“âˆž as xâ†’ +âˆž. So, the degree is even and the leading coefficient is negative.

ANSWER

degree: odd, leading

coefficient: negative

Guided Practice

Use synthetic substitution to evaluate the polynomial function for the given value of x.

6. f(x) = 5x3 + 3x2â€“x + 7;x = 2

7. g (x) = â€“2x4â€“x3 + 4x â€“ 5;x = â€“1

ANSWER

57

â€“10

ANSWER

8. Describe the degree and leading coefficient of thepolynomial function whose graph is shown.

To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.

a.

Example 5

Graph (a)f (x) = â€“x3 + x2 + 3x â€“ 3 and (b)f (x) = x4 â€“ x3 â€“ 4x2 + 4.

SOLUTION

The degree is odd and leading coefficient is negative. So, f (x) â†’ +âˆž asx â†’ â€“âˆž andf (x) â†’ â€“âˆž asx â†’ +âˆž.

b.

To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.

Example 5

The degree is even and leading coefficient is positive.

So, f (x) â†’ +âˆžasx â†’ â€“âˆžand f (x) â†’ +âˆžasx â†’ +âˆž.

Physical Science

The energy E (in foot-pounds) in each square foot of a wave is given by the model E = 0.0029s4 where sis the wind speed (in knots). Graph the model. Use the graph to estimate the wind speed needed to generate a wave with 1000 foot-pounds of energy per square foot.

Example 6

Example 6

SOLUTION

STEP 1

Make a table of values. The model onlydeals with positive values of s.

Plot the points and connect them with a smooth curve. Because the leading coefficient is positive and the degree is even, the graph rises to the right.

STEP 2

STEP 3

ï‚»

Examine the graph to see that s 24 when E = 1000.

The wind speed needed to generate the wave is about 24 knots.

ANSWER

Example 6

Guided Practice

Graph the polynomial function.

9. f(x) = x4 + 6x2 â€“ 3

10. f(x) = â€“x3 + x2 + x â€“ 1

Guided Practice

Graph the polynomial function.

11. f(x) = 4 â€“ 2x3

12.

WHAT IF? If wind speed is measured in miles per hour, the model in Example 6 becomes

E = 0.0051s4. Graph this model. What wind

speed is needed to generate a wave with 2000 foot-pounds of energy per square foot?

ANSWER

about 25 mi/h.

Guided Practice

ANSWER

no

71

ANSWER

Lesson Quiz

1. Tell whether the function f(x) = â€“3x3 â€“ 5xâ€“1 + 8is

a polynomial function. If so, write it in standard

form and state its degree, type, and leading

coefficient.

2. Use synthetic substitution to evaluate

f(x) =4x4 â€“ x3 + 3x2 + 5x â€“ 3when x =â€“2

ANSWER

45,560

Lesson Quiz

3. The estimated number of electric vehicles V in the

United States from 1995 to 2004 is given by the

equationV = 10x4 + 200x + 3000, wherexis the

number of years since 1995. Graph the model. Use

it to estimate the number of electric Vehicles in 2003.