Warm up

1

Determine algebraically whether each of the following functions is even, odd or neither.

Write the equation for transformation of.

Graphs of Polynomial Functions

Pre-Calculus

What You Should Learn

• Determine key features of a polynomial graph• Use the Leading Coefficient Test to determine

the end behavior of graphs of polynomial functions.

• Find and use zeros of polynomial functions as sketching aids.

• Find a polynomial equation given the zeros of the function.

Polynomials

• What do you remember about polynomials??• What would be key points of a polynomial?• Remember this …

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Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps.

Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

x

y

x

y

continuous not continuous continuoussmooth not smooth

polynomial not polynomial not polynomial

x

y f (x) = x3 – 5x2 + 4x + 4

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A polynomial function is a function of the form1

1 1 0( ) n nn nf x a x a x a x a

where n is a nonnegative integer and a1, a2, a3, … an are

real numbers. The polynomial function has a leading coefficient an and degree n.

Examples: Find the leading coefficient and degree of each polynomial function.

Polynomial Function Leading Coefficient Degree5 3( ) 2 3 5 1f x x x x

3 2( ) 6 7f x x x x

( ) 14f x

-2 5

1 3

14 0

Classification of a Polynomial

Degree Name Example

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Y = -2x5+3x4–x3+3x2–2x+6

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

constant Y = 3

linear Y = 5x + 4

quadratic Y = 2x2 + 3x - 2

cubic Y = 5x3 + 3x2 – x + 9

quartic Y = 3x4 – 2x3 + 8x2 – 6x + 5 quintic

Graphs of Polynomial Functions

The polynomial functions that have the simplest graphs are monomials of the form f (x) = xn, where n is an integer greater than zero.

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Polynomial functions of the form f (x) = x n, n 1 are called power functions.

If n is even, their graphs resemble the graph of

f (x) = x2.

If n is odd, their graphs resemble the graph of

f (x) = x3.

x

y

x

y

f (x) = x2

f (x) = x5

f (x) = x4

f (x) = x3

Moreover, the greater the value of n, the flatter the graph near the origin

The Leading Coefficient Test

Polynomial functions have a domain of all real numbers. Graphs eventually rise or fall without bound as x moves to the right.

Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test.

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Leading Coefficient TestAs x grows positively or negatively without bound, the value f (x) of the polynomial function

f (x) = anxn + an – 1x

n – 1 + … + a1x + a0 (an 0)

grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n

is odd or even.

x

y

x

y

n odd n even

an positive

an negative

Find the left and right behavior of the polynomial.

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823)( .4

132.3

14)(.2

23.1

4

52

34

35

xxxf

xxy

xxxf

xxxy

Zeros of Polynomial FunctionsIt can be shown that for a polynomial function f of degree n,

the following statements are true.

1. The function f has, at most, n real zeros.

2. The graph of f has, at most, n – 1 turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.)

Finding the zeros of polynomial functions is one of the most important problems in algebra.

Given the polynomials below, answer the following

A.What is the degree?B.What is its leading coefficient?C.How many “turns”(relative maximums or minimums) could it have

(maximum)?D.How many real zeros could it have (maximum)?E. How would you describe the left and right behavior of the graph of

the equation?F. What are its intercepts (y for all, x for 1 & 2 only)?

Equations:

14

144)(.4

123.3

82)(.2

23.1

6

25

24

23

xxxf

xxy

xxxf

xxxy

Warm- up

15 4 8 7 3

24 Krypto

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Warm- up

10 5 13 9 2

14 Krypto

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Zeros of Polynomial Functions

• There is a strong interplay between graphical and algebraic approaches to this problem.

• Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph.

• Finding zeros of polynomial functions is closely related to factoring and finding x-intercepts.

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A real number a is a zero of a function y = f (x)if and only if f (a) = 0.

A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa.

A polynomial function of degree n has at most n – 1 turning points and at most n zeros.

Real Zeros of Polynomial FunctionsIf y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.

1. a is a zero of f.2. a is a solution of the polynomial equation f (x) = 0.3. x – a is a factor of the polynomial f (x).

4. (a, 0) is an x-intercept of the graph of y = f (x).

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Example: Determine the multiplicity of the zeros of f (x) = (x – 2)3(x +1)4.

Zero Multiplicity Behavior

2

–1

3

4

odd

even

crosses x-axis at (2, 0) touches x-axis at (–1, 0)

Repeated ZerosIf k is the largest integer for which (x – a)

k is a factor of f (x)and k > 1, then a is a repeated zero of multiplicity k. 1. If k is odd the graph of f (x) crosses the x-axis at (a, 0). 2. If k is even the graph of f (x) touches, but does not cross through, the x-axis at (a, 0).

x

y

Example - Finding the Zeros of a Polynomial Function

Find all real zeros of

f (x) = –2x4 + 2x2.

Then determine the number of turning points of the graph of the function.

Example – Solution Solution:To find the real zeros of the function, set f (x) equal to zeroand solve for x.

–2x4 + 2x2 = 0 –2x2(x2 – 1) = 0

–2x2(x – 1)(x + 1) = 0

So, the real zeros are x = 0 (double root), x = 1, and x = –1.

Because the function is a fourth-degree polynomial, the graph of f can have at most 4 – 1 = 3 turning points.

Remove common monomial factor.

Factor completely.

cont’d

Set f (x) equal to 0.

Zeros of Polynomial Functions

In the example, note that because the exponent is greater than 1, the factor –2x2 yields the repeated zero x = 0.

Because the exponent is even, the graph touches the x-axis at x = 0.

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Another example: Find all the real zeros and turning points of the graph of f (x) = x

4 – x3 – 2x2.

Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).

The real zeros are x = –1, x = 0, and x = 2.

These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0).

The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.

y

x

f (x) = x4 – x3 – 2x2

Turning pointTurning point

Turning point

Zeros of Polynomial Functions

This means that when the real zeros of a polynomial function are put in order, they divide the real number line into intervals in which the function has no sign changes.

These resulting intervals are test intervals in which a representative x-value in the interval is chosen to determine if the value of the polynomial function is positive (the graph lies above the x-axis) or negative (the graph lies below the x-axis).

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Steps to Graphing Polynomials:

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Example: Sketch the graph of f (x) = 4x2 – x4.

Warm- up

21 8 24 725

2 Krypto

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Graph: f (x) x4 2x2 .

Let Review of what we do last class!!

Let Review of what we do last class!!

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After factoring, sketch the graph of the equation y = -x3+2x2-x

Find the Polynomial

Given the zeros, find an equation (assume lowest degree):Zeros: 2, 3Answer: (x – 2)(x – 3) = x2 – 5x +6Zeros: 0 (multiplicity of 2), -2, 5Answer: x2 (x + 2)(x – 5)= x2(x2 – 3x + 10)

= x4 – 3x3 + 10x2

Zeros: 2, 3 (multiplicity of 2), -4(multiplicity of 3) – leave in factored form

Answer: (x – 2)(x – 3)2 (x + 4)3

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Can you???

• Determine key features of a polynomial graph• Use the Leading Coefficient Test to determine

the end behavior of graphs of polynomial functions.

• Find and use zeros of polynomial functions as sketching aids.

• Find a polynomial equation given the zeros of the function.

Finding the Equation for a Polynomial Function by Hand.

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Finding the Equation for a Polynomial Function by Hand.

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•If the graph looks as though it just cuts cleanly through the x-axis, then the zero

has multiplicity one (see Figure(a)).

• If the graph looks like a quadratic and just touches the x-axis without cutting

through, then the zero has multiplicity two (see Figure(b)).

• If the graph looks like a cubic and has an inflection point as it cuts through the x-

axis, then the zero has multiplicity three (see Figure (c)).

Finding the Equation for a Polynomial Function by Hand.

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Finding the Equation for a Polynomial Function by Hand.

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Finding the Equation for a Polynomial Function by Hand.

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The idea here is to locate the x and y

coordinate of a point that is on the

graph of the polynomial function, but

which is not one of the zeros of the

polynomial function. The x and y are

substituted into the factored form,

allowing k to be found.

Example

• Write the equation of polynomial graph.

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Example

• Write the equation of polynomial graph.

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Time for worksheet

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