Evaluating a Polynomial Function

Evaluating Polynomials to Support Graphs

By Heather Eve

This lesson will review some skills and strategies that you should already know, and then you will be introduced to new skills and ideas.

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Please continue to add to your notes as you go through this lesson!​

Subject | Subject

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Review Questions

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Multiple Choice

What is the degree of this polynomial?

2x⁴ - 3x⁵ + x

• -3

• 2

• 4

• 5

Multiple Choice

What is the coefficient of this term?

9x³

• None

• 9

• 3

• x

Multiple Choice

Over the interval (−1, 2), the graph of the polynomial shown could be described as

• decreasing
• increasing
• quadratic
• complex

Multiple Choice

Which equation MOST LIKELY matches the graph?

• y = x⁴ - x³ + 3x² + 2
• y = -x⁴ + x³ + 5x + 2
• y = x³ - 2x² + 1x + 3
• y = -x³ - 2x² + 1x + 3

Review: Evaluating A Function

To "evaluate" a function is to analyze what happens at a specific point. For example, if you have an input, what is the output? Or vice versa.

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In common terms we simply say "Plug it in" Be mindful of which variable you are working with!​

Subject | Subject

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Review: Evaluating A Function

Don't forget! An exponent means you multiply the same number that many times.

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EXAMPLE:

y=x³​ for x=-3 --> (-3)(-3)(-3) =-27

Subject | Subject

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Multiple Choice

If f(x) = x²+ 3, find f(-2)

• -1

• 1

• 7

• -7

Fill in the Blank

Evaluate the polynomial.

$$f\left(n\right)=n^3-5n^2+9n-13\ \ \ \ \ \ at\ \ \ \ \ \ \ n=3$$  

Multiple Choice

Evaluate the polynomial.

$$f\left(a\right)=2a^3-3a^2-3a\ \ \ \ \ \ at\ \ \ \ \ a=2$$  

• -2

• 9

• -4

• -3

Multiple Choice

Evaluate the polynomial

$$f\left(m\right)=m^4+m^3-32m^2+3m+46\ \ \ \ \ at\ \ \ \ \ m=5$$  

• 88

• 150

• 11

• 25

Review: Zeroes Of A Function

Zeroes are places where ​y=0. You can find these by using any of the following strategies:

• Looking at a graph if available

• Factoring and setting each factor equal to 0

• Using the quadratic formula, for quadratics only

  (We will review more strategies specifically for polynomials soon!)​

Subject | Subject

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Review: Zeroes Of A Function

For a polynomial graph:

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If​ the function crosses through the x-axis it is a real zero.

If the function appears to bounce off of the x-axis, it is a real zero with a multiplicity of 2.

If the function appears to bounce in mid-air (turns around but not near the x-axis), there will be at least 2 imaginary zeroes.​

Subject | Subject

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Multiple Choice

What are the zeros of the function?

• 4,1
• 3,1,-2,-5
• -3, -1, 2, 5
• -2,4

Multiple Choice

Which root has a multiplicity of 2?

• -1

• 2

• 4

• -4

Multiple Choice

Identify the zeros

• x=2, x=6, x=-4
• (x+2)(x+6)(x-4)
• (x-2)(x-6)(x+4)
• x=-2, x=-6, x=4

Multiple Choice

What are the zeroes of the equation

y = (x - 2)(x + 4)

• x = -2 or x = 4

• x = 2 or x = -4

• x² - 2x - 8

• I don't know

Multiple Choice

What are the zeroes of the function

f(x) = x(x + 6)(x - 1)

• x = -6 or x = 1

• x = 6 or x = -1

• x = 0, x = -6 or x = 1

• x = 0, x = 6 or x = -1

Multiple Choice

If f(x) = 2x(x + 1)(3x - 4). What are the solutions to f(x) = 0?

• x = -1 or x = 4/3

• x = 0, x = -1, or x = 4/3

• x = 2, x = -1, or x = 3/4

• x = -1 or x = 4/3

Multiple Choice

What are the zeroes of this graph?

• x = -4, x = 0, x = 3, x =7

• x = -7, x = -3, x = 0, x = -4,

• (0,0)

• x = -4, x = 3, x =7

Multiple Choice

f(x) = (x+2)(x-3)²(x-4)

Which choice best describes the zeros?

• -2, 3 (multiplicity 2), 4

• -2 (multiplicity 2), 3, 4

• 2, -3 (multiplicity 2), -4

• 2 (multiplicity 2), -3, -4

NEW! Estimate a Max/Minimum Value

(Without needing a graph!)

In every polynomial, you can guarantee that between two zeroes--> as long as each zero has a multiplicity of one there will be​ a max or min somewhere in the middle. (if it has a multiplicity of two, AKA a bounce, then it IS a max or a min!)

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You can use an x value to evaluate the function and find the y value!​

Subject | Subject

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NEW! Estimate a Max/Minimum Value

(Without needing a graph!)

EXAMPLE:

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A polynomial has zeroes at -4, -2, 0, and 2.​

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• This means that a max/min must exist between -4 and -2, so plug in -3!​ This will get you on or near one of the extrema.

• A max/min must also exist between -2 and 0, so plug in -1!​ This will get you on or near another one of the extrema.​

• A max/min must also exist between 0 and 2, so plug in 1!​ This will get you on or near another one of the extrema.​​

Subject | Subject

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Multiple Choice

Which of the following points could possibly be a max or min for a polyomial with zeros: 9, 3, -2, 0

• (-6,0)

• (6,23)

• (3,4)

• (11,-2)

• (-4,-2)

Multiple Choice

Which of the following points could possibly be a max or min for this polynomial: f(x)= (x-3)(x+2)(x-5)

• (5,5)

• (-3,-48)

• (-2,0)

• (4,-6)

• (6,24)

Multiple Choice

Which of the following points could possibly be a max or min for this polynomial: f(x)= 2x³+5x²+4x+1 (it has two zeros at x=-1 and a zero at x=-0.5)

• (-0.5,0)

• (-1,0)

• (0,1)

• (1,12)

• (1,10)

Multiple Choice

Which of the following points could possibly be a max or min for this polynomial: f(x)= 6x³-32x²+6x+20 (part of it is shown on the graph)

• (5,0)

• (1,0)

• (0,1)

• (-1,-24)

• (3,-88)