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

– Quadratic polynomials

– Cubic polynomials

– Quartic polynomials

– Quintic polynomials

– X-intercepts

– Turning points

– Piece-wise functions

Constant Polynomials

• 𝑓 𝑥 = 𝑎

• Polynomial functions with degree of 0

• No x-Intercepts and no turning points

Linear Polynomials

• 𝑓 𝑥 = 𝑚𝑥 + 𝑏

• Also known as a linear function

• Polynomial function with degree of 1

• One x-intercept, no turning points

Quadratic Polynomials

• 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

• 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

• 𝑓 𝑥 = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑

• 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

• 𝑓 𝑥 = 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥2 + 𝑑𝑥 + 𝑘

• 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

• 𝑓 𝑥 = 𝑎𝑥5 + 𝑏𝑥4 + 𝑐𝑥3 + 𝑑𝑥2 + 𝑘𝑥 + 𝑙

• 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 a<0: Estimate x-intercepts: Determine minimum degree of f:

Determine the number of turning points: Identify turning points: State whether a>0 or a<0: Estimate x-intercepts: Determine minimum degree of f:

4 (-3,7) (-1.5,-1.5) (0,0) (1.5,-3) a>0 -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 𝑓(𝑥) at the given values of 𝑥:

𝑥 = -3, 1, 4

𝑓 𝑥 = 𝑥3 − 4𝑥2

3𝑥2

𝑥3 − 54

𝑖𝑓𝑖𝑓𝑖𝑓

𝑥 ≤ −3−3 < 𝑥 < 4

𝑥 ≥ 4

𝑓 −3 = (−3)3−4 −3 2 = −63

𝑓 1 = 3(1)2= 3 𝑓 4 = (4)3−54 = 10

Piece-wise functions continued:

• Graph:

𝑓 𝑥 = 2𝑥−2

𝑥2 − 4

𝑖𝑓𝑖𝑓𝑖𝑓

−5 ≤ 𝑥 < −1−1 ≤ 𝑥 < 00 ≤ 𝑥 ≤ 2

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