Sullivan Algebra and Trigonometry: Section 5.1 Polynomial Functions

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Sullivan Algebra and Trigonometry: Section 5.1 Polynomial Functions. Objectives Identify Polynomials and Their Degree Graph Polynomial Functions Using Transformations Identify the Zeros of a Polynomial and Their Multiplicity Analyze the Graph of a Polynomial Function.

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Presentation Transcript
• Objectives
• Identify Polynomials and Their Degree
• Graph Polynomial Functions Using Transformations
• Identify the Zeros of a Polynomial and Their Multiplicity
• Analyze the Graph of a Polynomial Function

A polynomial function is a function of the form

where an , an-1 ,…, a1 , a0are real numbers and n is a nonnegative integer. The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears.

A power function of degree n is a function of the form

where a is a real number, a 0, and n > 0 is an integer.

Summary of Power Functions with Even Degree

1) Symmetric with respect to the y-axis.

2) Domain is the set of all real numbers. Range is the set of nonnegative real numbers.

3) The graph always contains the points (0, 0); (1, 1); and (–1, 1).

4) As the exponent increases in magnitude, the graph increases very rapidly as x increases, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

(-1, 1)

(1, 1)

(0, 0)

Summary of Power Functions with Odd Degree

1) Symmetric with respect to the origin.

2) Domain is the set of all real numbers. Range is the set of all real numbers.

3) The graph always contains the points (0, 0); (1, 1); and (–1, –1).

4) As the exponent increases in magnitude, the graph becomes more vertical when x > 1 or x < –1, but for –1 < x < 1, the graphs tends to flatten out and lie closer to the x-axis.

(1, 1)

(0, 0)

(-1, -1)

(1,1)

(0,0)

(0,0)

(1, -2)

Graph the following function using transformations.

(1, 4)

(1,0)

(2, 2)

(2,-2)

2

f

(

x

)

=

(

x

+

1

)(

x

-

4

)

Consider the polynomial:

2

f

(

x

)

=

(

x

+

1

)(

x

-

4

)

= 0

Solve the equation f (x) = 0.

x + 1 = 0 OR x– 4 = 0

x = –1 OR x = 4 (–1, 0) and (4, 0) are the x-intercepts)

If f is a polynomial function and r is a real number for which f (r) = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then

a) (r, 0) is an x-intercept of the graph of f.

b) (x–r) is a factor of f.

If is a factor of a polynomial f and is not a factor of f, then r is called a zero of multiplicity m of f.

Example: Find all real zeros of the following function and their multiplicity.

x = 3 is a zero with multiplicity 2.

x = –7 is a zero with multiplicity 1.

x = 1/2 is a zero with multiplicity 5.

If r is a Zero of Odd Multiplicity

Sign of f (x) changes from one side to the other side of r.

Graph crosses

x-axis at r.

If r is a Zero of EvenMultiplicity

Sign of f (x) does not change from one side to the other side of r.

Graph touches

x-axis at r.

Theorem: For large values of x, either positive or negative, the graph of the polynomial

resembles the graph of the power function

2

f

(

x

)

=

x

+

1

x

-

5

x

+

4

For the polynomial

(

)

(

)

(

)

a) Find the x- and y-intercepts of the graph of f.

The x-intercepts (zeros) are (–1, 0), (5,0), and (–4,0).

To find the y-intercept, evaluate f(0).

So, the y-intercept is (0, –20)

2

f

(

x

)

=

x

+

1

x

-

5

x

+

4

For the polynomial

(

)

(

)

(

)

b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

x = –4 is a zero of multiplicity 1 (crosses the x-axis)

x = –1 is a zero of multiplicity 2 (touches the x-axis)

x = 5 is a zero of multiplicity 1 (crosses the x-axis)

c) Find the power function that the graph of f resembles for large values of x.

2

f

(

x

)

=

x

+

1

x

-

5

x

+

4

For the polynomial

(

)

(

)

(

)

(6, 490)

(-1, 0)

(-5, 160)

(0, -20)

(5, 0)

(-4, 0)

(-2, -14)