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Section 4.3 – The Graphs of Polynomial Functions

Objectives

• Identify polynomial functions. • Sketch graphs of power functions. • Determine the end behavior of polynomials from the leading term property. • Given the graph of a polynomial, determine the possible degree of the polynomial, the

constant coefficient, and sign of the leading coefficient. • Determine the intercepts of the graph of a polynomial function. • Given a polynomial function in factored form, determine the zeros and their

multiplicities. • Sketch the graph of a polynomial function. • Determine a possible equation of a polynomial function given its graph.

Preliminaries

A polynomial function is a function of the form: 𝑓(𝑥) = where 𝑎0, 𝑎1, 𝑎2, … , 𝑎𝑛 are real numbers and n is a non-negative integer. Examples? Terms associated with a polynomial The degree of the polynomial is .

𝑎0, 𝑎1, 𝑎2, ⋯ , 𝑎𝑛 are called the .

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𝑎𝑛 is called the .

𝑎𝑛𝑥𝑛 is called the .

𝑎0 is called the .

The domain of every polynomial function is .

Example

For the polynomial function 𝑓(𝑥) = −7𝑥5 + 19𝑥4 + 3𝑥2 −2

5𝑥 − 18, determine each of the

following.

Degree:

Leading coefficient:

Leading term:

Constant term:

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Warm-up 5. Determine the transformations that are performed on a base function and sketch a

graph of the given function.

(A) 𝑎(𝑥) = −𝑥2 + 5

(B) 𝑐(𝑥) = (𝑥 − 4)3 − 1

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6. Sketch the graphs of 𝑦 = 𝑥𝑛 for 𝑛 = 1, 2, 3, 4, 5, and 6.

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

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Class Notes and Examples 4.3.1 Determine whether each of the following represents a polynomial function. If the

function is a polynomial, state the degree and leading coefficient. If the function is not a polynomial, explain why.

(A) 𝑃(𝑥) = 4𝑥2 − 3𝑥−1

(B) 𝑄(𝑥) = √5𝑥4 − 4𝑥3 + 6

(C) 𝑅(𝑥) =7+2𝑥2−3𝑥4

5

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How can you determine the end behavior of a polynomial function of the form 𝑓(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + 𝑎1𝑥 + 𝑎0 ?

Sketch the four possible end behaviors for a polynomial function below. Also, describe what is happening to y as x approaches −∞ and +∞.

Even degree Positive leading coefficient

Odd degree Positive leading coefficient

Even degree Negative leading coefficient

Odd degree Negative leading coefficient

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4.3.2 Consider the following graphs of polynomial functions. Determine whether the leading coefficient is positive or negative and whether the degree is even or odd.

(A)

(B)

-20

-16

-12

-8

-4

4

8

12

-4 -3 -2 -1 1 2 3 4

-12

-8

-4

4

8

12

16

20

-4 -3 -2 -1 1 2 3 4

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Does every polynomial function have a 𝑦-intercept? How do you find it?

How do you find the zeros of a polynomial function?

4.3.3 Determine the zeros of each polynomial by factoring.

(A) 𝑓(𝑥) = 𝑥3 − 5𝑥2 − 4𝑥 + 20

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(B) 𝑁(𝑡) = 𝑡4 − 18𝑡2 + 81

What is meant by the multiplicity of a zero? In other words, if c is a real zero of a polynomial function f, then 𝑥 = 𝑐 is a zero of multiplicity k if:

What are the zeros and their multiplicities for the polynomial function given below? 𝑓(𝑥) = 𝑥2(𝑥 + 5)(𝑥 − 2)3(2𝑥 − 7)4

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Note: There are 3 general cases for the shape of the graph of a polynomial near a zero 𝑥 = 𝑐. When 𝑥 = 𝑐 is a zero with multiplicity 1, then the shape of the graph near 𝑥 = 𝑐 looks linear, and appears as either:

OR c c When 𝑥 = 𝑐 is a zero with even multiplicity (≥ 𝟐), then the shape of the graph near 𝑥 = 𝑐 looks like a parabola, and appears as either:

OR c c When 𝑥 = 𝑐 is a zero with odd multiplicity (≥ 𝟑), then the shape of the graph near 𝑥 = 𝑐 looks like a cubic, and appears as either:

OR c c

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If a polynomial has degree n, how many turning points can it possibly have? 4.3.4 For the polynomial functions graphed below, determine the sign of the leading

coefficient and minimum possible degree. (A)

(B)

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How can we graph a polynomial function by hand?

4.3.5 Determine the zeros and their multiplicities for the following polynomial functions.

Using the information about end behavior, zeros, and multiplicities, sketch a graph of each by hand.

(A) 𝑀(𝑥) = 𝑥3 − 4𝑥

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(B) 𝑅(𝑥) = 𝑥(𝑥 + 3)2(𝑥 − 2)3 (C) 𝑓(𝑥) = 𝑥2(2𝑥 + 3)(𝑥 − 4)3

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-6

-4

-2

2

4

6

-4 -3 -2 -1 1 2 3 4

(D) 𝑔(𝑥) = (𝑥2 + 1)(3𝑥 − 5)2(𝑥 + 1)

4.3.6 Determine a possible equation for the polynomial functions graphed below. Verify by

graphing on your calculator. (A)

(0, −2)

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-8

-6

-4

-2

2

4

6

8

-4 -3 -2 -1 1 2 3 4

-8

-6

-4

-2

2

4

6

8

-4 -3 -2 -1 1 2 3 4

(B) (0, 6)

(C) (1, 1.5)

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-2

-1

1

2

3

4

5

6

-1 1 2 3 4 5 6

(D) (0, 5)

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-5

-4

-3

-2

-1

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

Section 4.3 Self-Assessment (Answers on page 256) 1. (Multiple Choice) What is the degree of the following polynomial function?

𝑓(𝑥) =1

3(𝑥 + 14)4(𝑥 − 3)2(𝑥 − 9)

(A) 1

3 (B) 4 (C) 6 (D) 7 (E) None of these

2. (Multiple Choice) The polynomial function 𝑇(𝑥) = 5(𝑥2 + 5)(𝑥 − 5)3(𝑥 + 5)4 has a

zero at 𝑥 = 5. What is the multiplicity of this zero?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 9 3. Determine a possible equation for the polynomial function graphed below.

(2, 1)

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4. (Multiple Choice) Determine the end behavior of the graph of the polynomial function 𝑃(𝑥) = −3(𝑥 + 2)(𝑥 − 14)3(𝑥 + 15)2.

(A) 𝑃(𝑥) approaches −∞ as 𝑥 approaches −∞

𝑃(𝑥) approaches ∞ as 𝑥 approaches ∞

(B) 𝑃(𝑥) approaches ∞ as 𝑥 approaches −∞ 𝑃(𝑥) approaches ∞ as 𝑥 approaches ∞

(C) 𝑃(𝑥) approaches ∞ as 𝑥 approaches −∞

𝑃(𝑥) approaches −∞ as 𝑥 approaches ∞

(D) 𝑃(𝑥) approaches −∞ as 𝑥 approaches −∞ 𝑃(𝑥) approaches −∞ as 𝑥 approaches ∞

5. Determine the equation for the polynomial function in factored form that has the

following characteristics.

Zeros at 𝑥 = −4 (multiplicity 2) and 𝑥 = 2 (multiplicity 3)

Passes through the point (0, 16)