4.3 Graphs of Polynomials

Multiplicity, End Behavior and Graphing Polynomials

Draw

Bellwork:

Please complete the chart: (use your notes!)

Use the chart to complete your notes for Summary of Graphing Parent Functions.​

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Polynomials - Graphing

End Behavior

​Notation

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If the leading coefficient is positive then the graph will always end up (right leg)

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If the leading coefficient is negative the graph will always end going down (left leg).

Multiple Choice

The end behavior of a polynomial function is determined by the DEGREE and the SIGN (+/-) of the Leading Coefficient (LC). Identify the DEGREE of the polynomial and the SIGN(+/-) of the LC

• LC = +

  Degree = Even

• LC = +

  Degree = Odd

• LC = -

  Degree = Even

• LC = -

  Degree = Odd

Multiple Choice

The end behavior of a polynomial function is determined by the DEGREE and the SIGN (+/-) of the Leading Coefficient (LC). Identify the DEGREE of the polynomial and the SIGN(+/-) of the LC

• LC = +

  Degree = Even

• LC = +

  Degree = Odd

• LC = -

  Degree = Even

• LC = -

  Degree = Odd

Multiple Choice

The end behavior of a polynomial function is determined by the DEGREE and the SIGN (+/-) of the Leading Coefficient (LC). Identify the DEGREE of the polynomial and the SIGN(+/-) of the LC

• LC = +

  Degree = Even

• LC = +

  Degree = Odd

• LC = -

  Degree = Even

• LC = -

  Degree = Odd

Multiple Choice

The end behavior of a polynomial function is determined by the DEGREE and the SIGN (+/-) of the Leading Coefficient (LC). Identify the DEGREE of the polynomial and the SIGN(+/-) of the LC

• LC = +

  Degree = Even

• LC = +

  Degree = Odd

• LC = -

  Degree = Even

• LC = -

  Degree = Odd

Factored form of a Polynomial

By taking a closer look at the factors we can determine the multiplicity of each factor and its corresponding root

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​Looking at the factors

determines the behavior of the graph at the zeros.

​Multiplicity

An even (2, 4, 6, 8, ...) multiplicity indicates that a zero is a turning point in the graph.


An odd (1, 3, 5, 7, ...) multiplicity indicates that the the graph will cross over the x-axis at the zero.

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Review of Graphs of Polynomial Functions

Keep track of your answers so you can provide them when needed

Part 1: Review of end behavior and factoring

When in doubt... graph it!

• What is the parity of the degree?  (odd or even?)

• What sign is the lead coefficient? (positive or negative?)

Multiple Choice

Given the function

$$f\left(x\right)=x^3-4x^2+3x-12$$  What is its end behavior?

• Left, down; right, down

• Left, up; right, up

• Left, down; right, up

• Left, up; right, down

• What is its factored form?

• Take time to work on it on your own first.

Multiple Choice

Given the function

$$f\left(x\right)=x^3-4x^2+3x-12$$  What is its factored form?

• It cannot be factored

• $$\left(x^2-4\right)\left(x+3\right)$$  
  

• $$\left(x^2-3\right)\left(x+4\right)$$  
  

• $$\left(x^2+3\right)\left(x-4\right)$$

Part 2: Review of zeros, multiplicity, and behavior at x-intercepts.

Pay close attention to exponents!

• What are the zeros?

• What is their multiplicity? (odd or even?)

• Write it down on your paper first.

Multiple Choice

Given the function

$$f\left(x\right)=-x\left(x+2\right)\left(x-5\right)^2$$  What are the zeros?  What is their multiplicity?

• 0, even; -2, odd; 5, odd

• -2, odd; 5, even

• 0, odd; -2, odd; 5, even

• 0, odd; -2, even; 5, odd

• Where are the x-intercepts?

• What is the behavior of the graph at each x-intercept?

Multiple Choice

Given the function

$$f\left(x\right)=-x\left(x+2\right)\left(x-5\right)^2$$  Where are the x-intercepts?  What is the behavior of the graph at each x-intercept?

• (0,0); (-2,0); (5,0):  Cross, bounce, cross

• (0,0); (-2,0); (5,0):  Bounce, bounce, cross
  

• (0,0); (-2,0); (5,0):  Cross, bounce, bounce
  

• (0,0); (-2,0); (5,0):  Cross, cross, bounce
  

Check your work!

Part 3: Review of relative extrema and domains of increase or decrease

When in doubt... graph it!

How many relative extrema does this graph have?  (Think: "How many times does it turn?")

Multiple Choice

Given the function

$$f\left(x\right)=-x\left(x+2\right)\left(x-5\right)^2$$  How many relative extrema are there?  What type?

• Three:  two relative minimums and one relative maximum

• Three:  two relative maximums and one relative minimum
  

• Two:  one relative maximum and one relative minimum

• One:  relative minimum