Sketching Higher Order Polynomials

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What is the DEGREE of the polynomial

$f\left(x\right)=3x^4+2x^3-7x^2-5x+9$  

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What is the LEADING COEFFICIENT of the polynomial

$f\left(x\right)=3x^4+2x^3-7x^2-5x+9$  

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What is the DEGREE of the polynomial

$g\left(x\right)=-4x^3-8x^2+2x-10$  

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What is the LEADING COEFFICIENT of the polynomial

$g\left(x\right)=-4x^3-8x^2+2x-10$  

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What is the DEGREE of the polynomial

$h\left(x\right)=6x^4+x^5-3x^2+4x$  

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What is the LEADING COEFFICIENT of the polynomial

$h\left(x\right)=6x^4+x^5-3x^2+4x$  

Think Quadratics

​Concave up vs. Concave down

​Think Linear

​Positive slope vs. Negative slope

​Summary of End Behaviors

Multiple Choice

Does the graph show an EVEN or ODD degree polynomial?

• EVEN degree

• ODD degree

Multiple Choice

Does the graph have a POSITIVE or NEGATIVE leading coefficient?

• POSITIVE

• NEGATIVE

Multiple Choice

Does the graph show an EVEN or ODD degree polynomial?

• EVEN degree

• ODD degree

Multiple Choice

Does the graph have a POSITIVE or NEGATIVE leading coefficient?

• POSITIVE

• NEGATIVE

Multiple Choice

Does the graph show an EVEN or ODD degree polynomial?

• EVEN degree

• ODD degree

Multiple Choice

Does the graph have a POSITIVE or NEGATIVE leading coefficient?

• POSITIVE

• NEGATIVE

Multiple Select

Which functions' graph would have a decreasing left end behavior and increasing right end behavior?

NOTE: There may be more than one correct answer.

• $$f\left(x\right)=2x^3+4x^2+6x+3$$  

• $$g\left(x\right)=-8x^3+2x^2+4x$$  

• $$h\left(x\right)=-4x^2+6x^3+x+2$$  

Multiple Choice

Which of the function's graph would have a INCREASING left end behavior and INCREASING right end behavior?

• $$f\left(x\right)=-4x^2+6x+3$$  

• $$g\left(x\right)=8x^3+2x^2+4x$$  

• $$h\left(x\right)=2x^4+6x^3+x+2$$  

Multiple Choice

Which graphs would have a NEGATIVE leading coefficient?

Open Ended

What do you notice/wonder about the cubic functions and their graphs? How are they similar? How are they different?

​Multiplicity

• A zero has MULTIPLICITY.

• ​Multiplicity is the number of times the zero's associated factor appears in the polynomials.

• ​You can determine if the graph will bounce or cross  the x-axis from the multiplicity  

  • ​If the multiplicity is odd, the graph will cross at that zero

  • ​If the multiplicity is even, the graph will bounce at that zero 

​Theorem: Turning Points

• If f is a polynomial function of degree n, then the graph of f has at most n-1 turning points.

• If the graph of a polynomial function f has n-1 turning points, the the degree of f is a least n.

​

The graph on the right has 3 turning points so that means the degree of the function is at least 4.

Multiple Select

Select all the zeros (with multiplicities) of

$$f\left(x\right)=\left(x+2\right)\left(x-1\right)^2$$  

• x= -2 multiplicity 2

• x= 2 multiplicity 1

• x= -2 multiplicity 1

• x= -1 multiplicity 2

• x= 1 multiplicity 2

Multiple Select

$$f\left(x\right)=\left(x+2\right)\left(x-1\right)^2$$  

From the last question, we know that f(x) has zeros at

x= -2 multiplicity 1 and x= 1 multiplicity 2.

Select whether the graph of f(x) will cross or bounce at each of the zeros.

• cross at x= -2

• bounce at x= -2

• cross at x=1

• bounce at x=1

Multiple Choice

Which of the following is the graph of

$$f\left(x\right)=\left(x+2\right)\left(x-1\right)^2$$  ?

Multiple Select

Select all the zeros (with multiplicities) of

$$g\left(x\right)=-2x^2\left(x+4\right)^2$$  

• x= -4 multiplicity 2

• x= 4 multiplicity 2

• x= 0 multiplicity 2

• x= 2 multiplicity 2

• x= -2 multiplicity 2

Multiple Select

$$g\left(x\right)=-2x^2\left(x+4\right)^2$$  

From the last question, we know that g(x) has zeros at

x= 0 multiplicity 2 and x= -4 multiplicity 2.

Select whether the graph of g(x) will cross or bounce at each of the zeros.

• cross at x= 0

• bounce at x= 0

• cross at x= -4

• bounce at x= -4

Multiple Choice

Which of the following is the graph of

$$g\left(x\right)=-2x^2\left(x+4\right)^2$$  ?

Multiple Select

x=0 multiplicity of 1 is a zero of h(x).

Select all other zeros (with multiplicities) of

$$h\left(x\right)=x\left(x-7\right)^2\left(x+5\right)^2$$  

• x= -7 multiplicity 2

• x= 7 multiplicity 2

• x= -5 multiplicity 2

• x= 5 multiplicity 2

Multiple Choice

$$h\left(x\right)=x\left(x-7\right)^2\left(x+5\right)^2$$  

From the last question, we know that h(x) has zeros at

x= 0 multiplicity 1, x= 7 multiplicity 2, and x= -5 multiplicity 2

Select the graph of h(x).

Multiple Select

Select all other zeros (with multiplicities) of

$$k\left(x\right)=-\left(x+1\right)^2\left(x+4\right)$$  

• x= 0 multiplicity 1

• x= -1 multiplicity 2

• x= 1 multiplicity 2

• x= -4 multiplicity 1

• x=4 multiplicity 1

Multiple Choice

$$k\left(x\right)=-\left(x+1\right)^2\left(x+4\right)$$  

From the last question, we know that k(x) has zeros at

x= -1 multiplicity 2, and x= -4 multiplicity 1

Select the graph of h(x).

Multiple Select

Select ALL the possible factors in the equation of the graph.

• $$\left(x-2\right)$$  

• $$\left(x+1\right)$$  

• $$\left(x-1\right)^2$$  

• $$\left(x+1\right)^2$$  

• $$\left(x+2\right)$$  

​Writing Polynomial Functions

Use smallest degree possible​

Open Ended

Which of these doesn't belong? Explain your choice.