Unit 7 Test Review: Polynomial Functions

Unit 7 Test Review: Polynomial Functions

by Hannah Wiley

​Standards:

​A2.AAPR.3: Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.

(Limit to polynomials with degrees 3 or less.)

​

Unit 7 Covered...

• Degree of Polynomial Functions

• ​Naming Polynomials by Degree & Number of Terms

• ​Shape of Polynomials

• ​End Behavior

• ​Increasing/Decreasing Intervals

• ​Extrema of Polynomials

• ​Zeros of Polynomials

• ​Multiplicity

​Degree of Polynomial Functions

• ​The degree of a polynomial is the highest degree (largest exponent) of its terms.

• ​Sometimes, we need to add together the exponents of the factors that are given.

• ​Even degree polynomials end behavior point in the same direction.

• ​Odd degree polynomials end behavior point in opposite directions.

Multiple Choice

1) Degree of 2x+5 is ____________

• 2

• 1

• 5

• 4

Multiple Choice

3) Degree of

$$5a^4+7a^2-3a^9+10$$  is ____

• 2

• 4

• 9

• 10

Multiple Choice

9) Degree of a Quadratic polynomial is _______

• 1

• 2

• 3

• 0

Multiple Choice

What is the degree of P(x) = (x + 5)(x - 3)(x + 2)

• 3

• 2

• 1

Multiple Choice

What is the degree of P(x) = (x)(x - 4)

• 0

• 1

• 2

​Naming Polynomials by # of Terms & Degree

• ​Number of terms in a polynomial is given by the terms that are added or subtracted together.

  • ​Monomial: 1 term

  • ​Binomial: 2 terms

  • ​Trinomial: 3 terms

  • ​Polynomial: More than 3 terms

• ​Certain degrees of polynomials have special names.

  • ​1st Degree: Linear

  • ​2nd Degree: Quadratic

  • ​3rd Degree: Cubic

  • ​4th Degree: Quartic

  • ​5th Degree: Quintic

Multiple Choice

Classify:
2x⁴+14x³−2x²

• Monomial
• Binomial
• Trinomial
• Polynomial

Multiple Choice

Classify:
2n³

• Monomial
• Binomial
• Trinomial
• Polynomial

Multiple Choice

Classify:
x³ + 3x

• Monomial
• Binomial
• Trinomial
• Polynomial

Multiple Choice

Classify:
6x³+3x²+2x−4

• Monomial
• Binomial
• Trinomial
• Polynomial

Multiple Choice

Classify by degree of polynomial:
-2x² – x

• Constant
• Linear
• Quadratic
• Cubic

Multiple Choice

Classify by degree of polynomial:
9x

• Constant
• Linear
• Quadratic
• Cubic

Multiple Choice

Classify by degree of polynomial:
7x³ – 8x² -4x+ 9

• Constant
• Linear
• Quadratic
• Cubic

​Shape of Polynomials

• ​Even degree Polynomials have matching end behaviors, either both going up or both going down.

  • ​when the leading coefficient is POSITIVE, both ends point UP

  • ​when the leading coefficient is NEGATIVE, both ends point DOWN

• ​Odd degree Polynomials have opposite end behaviors, one going up and one going down.

  • ​when the leading coefficient is POSITIVE, the graph goes UP from left to right

  • ​when the leading coefficient is NEGATIVE, the graph goes DOWN from left to right

Multiple Choice

Describe the degree and leading coefficient for the polynomial shown.

• Lead Coefficient: Positive

  Degree: Even

• Lead Coefficient: Positive

  Degree: Odd

• Lead Coefficient: Negative

  Degree: Even

• Lead Coefficient: Negative

  Degree: Odd

Multiple Choice

Describe the degree and leading coefficient for the polynomial shown.

• Lead Coefficient: Positive

  Degree: Even

• Lead Coefficient: Positive

  Degree: Odd

• Lead Coefficient: Negative

  Degree: Even

• Lead Coefficient: Negative

  Degree: Odd

Multiple Choice

Describe the degree and leading coefficient for the polynomial shown.

• Lead Coefficient: Positive

  Degree: Even

• Lead Coefficient: Positive

  Degree: Odd

• Lead Coefficient: Negative

  Degree: Even

• Lead Coefficient: Negative

  Degree: Odd

Multiple Choice

Describe the degree and leading coefficient for the polynomial shown.

• Lead Coefficient: Positive

  Degree: Even

• Lead Coefficient: Positive

  Degree: Odd

• Lead Coefficient: Negative

  Degree: Even

• Lead Coefficient: Negative

  Degree: Odd

​End Behavior

• ​End behavior reads the graph as it goes to the left (-∞) and as it goes to the right (∞)

Multiple Choice

Describe the end behavior of the graph.

• x → ∞, y→ ∞ and
  x→ ∞, y→⁻∞
• x → ∞, y→ ∞ and
  x→⁻∞, y→∞
• None of these
• x →∞,y→⁻∞ and
  x→∞, y→⁻∞

Multiple Choice

Complete the end behavior statement for the graph of f(x). As  $$x\rightarrow\infty,\ y\rightarrow\ ?$$  

• $$\infty$$  

• $$-\infty$$  

Multiple Choice

Complete the end behavior statement for the graph of f(x). As  $$x\rightarrow-\infty,\ y\rightarrow\ ?$$  

• $$\infty$$  

• $$-\infty$$  

​Increasing/Decreasing Intervals

• ​When a function is going up, from left to right, the function is increasing

• ​When a function is going down, from left to right, the function is decreasing

• ​To make the interval, read the values on the x-axis where the graph is increasing or decreasing between.

Multiple Choice

What is the increasing interval on the function shown?

• $$\left(-\infty,\ 1\right)$$

• $$\left(-\infty,\ 2\right)$$

• $$\left(2,\ \infty\right)$$

• $$\left(1,\ \infty\right)$$

Multiple Choice

What is the decreasing interval on the function shown?

• $$\left(-\infty,\ 1\right)$$

• $$\left(-\infty,\ 2\right)$$

• $$\left(2,\ \infty\right)$$

• $$\left(1,\ \infty\right)$$

Multiple Choice

What is the increasing interval(s) on the function shown?

• $$\left(-\infty,\ -15\right)\ and\ \left(9,\ -15\right)$$

• $$\left(-2,\ 9\right)\ and\ \left(2,\ \infty\right)$$

• $$\left(-\infty,\ -2\right)\ and\ \left(0,\ 2\right)$$

• $$\left(-2,\ 0\right)\ and\ \left(2,\ \infty\right)$$

Multiple Choice

What is the decreasing interval(s) on the function shown?

• $$\left(-\infty,\ -15\right)\ and\ \left(9,\ -15\right)$$

• $$\left(-2,\ 9\right)\ and\ \left(2,\ \infty\right)$$

• $$\left(-\infty,\ -2\right)\ and\ \left(0,\ 2\right)$$

• $$\left(-2,\ 0\right)\ and\ \left(2,\ \infty\right)$$

​Extrema of Polynomials

• ​Extrema of Polynomials are the minimum values and the maximum values.

  • ​Minimum Values: the lowest values in the functions

    • ​Relative/Local Minimum: the lowest value in a section

    • ​Absolute Minimum: the lowest value in the entire function

  • ​Maximum Values: the highest values in the functions

    • ​Relative/Local Minimum: the highest value in a section

    • ​Absolute Maximum: the highest value in the entire function

Multiple Choice

What is the location of the absolute maximum for the graphed function?

• x = 1

• x = 3

• x = 4

• x = 5

• Does Not Exist

Multiple Choice

What is the location of the relative maximum within the given interval of the graphed function?

$$-5\le x\le0$$  

• x = -4

• x = -2

• x = 0

• x = 3

• Does Not Exist

Multiple Choice

What are the extrema of the graph?

• Max = (2, -3), (1, 2)

  Min = (-3, -1), (-1, 4)

• Max = (-3, -1), (-1, 4)

  Min = (2, -3), (1, 2)

• Max = (-1, -3), (4, -1)

  Min = (-3, 2), (2, 1)

• Max = (-3, 2), (2, 1)

  Min = (-1, -3), (4, -1)

​Zeros of Polynomial Function

• ​Zeros of Polynomial Functions are where the graph crosses the x-axis.

  ​Zeros are also where the Polynomial Functions equal zeros.

  • ​when given the factors of a function, set each factor equal to zero and solve for the variable.

Multiple Choice

Find the zeros of the polynomial.
f(x) = (x-5)(2x+3)(7x-4)(x+6)

• x = 5, (2/3), 6, (7/4)
• x= -5, (3/2), (-4/7), 6
• x= 5, (-3/2), (4/7), -6
• x= 6, 5, -2/3, -4/7

Multiple Choice

For $$\left(x+5\right)\left(x-1\right)=0$$  , what is x equal to? (You can only select 1 answer!!!) 

• x=-5

• x=-4 x=1

• x=1

• x=1 and x=-5

• x=-1 and x=5

Multiple Choice

If the zeros of a polynomial are -1, 3, and 7 then what does the polynomial look like in factored form?

• (x-1)(x+3)(x+7)

• (x+1)(x-3)(x-7)

• -x² + 3x + 7

• (x-1)

​Multiplicity of Factors

• The multiplicity of a factor is the number of times it is multiplied times itself.

• ​Even multiplicity means that the graph touches the x-axis.

• ​Odd multiplicity means that the graph crosses the x-axis.

Multiple Choice

If the graph of a function crosses the x-axis, what does that mean about the multiplicity of that zero?

• Even

• Odd

• None

Multiple Choice

If the there is a turning point on the x-axis, what does that mean about the multiplicity of that zero?

• Even

• Odd

• None