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### Transcript of Section 4.2 Review

• College Algebra

Chapter 4, section 2 Created by Lauren Atkinson

Mary Stangler Center for Academic Success

This review is meant to highlight basic concepts from Chapter 4. It does not cover all concepts presented by your instructor. Refer back to your notes, handouts, the book, MyMathLab, etc. for further prepare for your exam.

• 4.2: Polynomial Functions and Models

This section covers the following: Constant polynomials

Linear polynomials

Cubic polynomials

Quartic polynomials

Quintic polynomials

X-intercepts

Turning points

Piece-wise functions

• Constant Polynomials

=

Polynomial functions with degree of 0

No x-Intercepts and no turning points

• Linear Polynomials

= +

Also known as a linear function

Polynomial function with degree of 1

One x-intercept, no turning points

= 2 + +

Polynomial functions with a degree of 2

Can have 0,1 or 2 x-intercepts; one turning point (turning point is the vertex)

• Cubic polynomials

= 3 + 2 + +

Polynomial function with a degree of 3

End behavior: falls to the left, rises to the right OR falls to the right, rises to the left

Can have 0 or 2 turning points; can have 1,2 or 3 x-intercepts

Must cross the x-axis at least once

• Quartic polynomials

= 4 + 3 + 2 + +

Polynomial function with degree of 4

End behavior: falls to the left and falls to the right OR rises to the left and rises to the right

Can have 1,2 or 3 turning points; can have 0,1,2,3 or 4 x-intercepts

• Quintic polynomials

= 5 + 4 + 3 + 2 + +

Polynomial function with degree of 5

End behavior: falls to the left and rises to the right OR falls to the right and rises to the left

Can have 0,1,2,3 or 4 turning points; can have 0,1,2,3,4 or 5 x-intercepts

• Constant Linear Quadratic Cubic Quartic Quintic

• X-intercepts

Occur when the graph hits or touches the x-axis: also called a zero

Polynomials with varying degrees have at most that number of x-intercepts [this depends on their degree] For example: a function with degree 27 can have anywhere from 0 to 27 x-intercepts

• Turning Points

Occur when the slope changes from positive to negative or negative to positive

They occur at either the peak or the valley of the graph [this is also where maximums and minimums occur]

The number of turning points= at most 1 less than the degree of

the polynomial. For example: a polynomial of degree 14 can have from 0 to 13 turning points AND a polynomial with 13 turning points has to be a polynomial of degree 14

• Examples:

Determine the number of turning points: Identify turning points: State whether a>0 or a0 or a0 -4,-2,0,2 5

3 (-0.5,-1) (0.5,1) (-2,-7) a>0 -1,0,1,3 4

• Piece-wise functions

Learn through examples:

Evaluate () at the given values of :

= -3, 1, 4

= 3 42

32

3 54

33 < < 4

4

3 = (3)34 3 2 = 63

1 = 3(1)2= 3 4 = (4)354 = 10

• Piece-wise functions continued:

Graph:

= 22

2 4

5 < 11 < 00 2

• This is generally how the piece-wise function from the previous slide should look: