Unit 6: Polynomial Functions

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Mathematics

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9th - 12th Grade

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Natalie McIntosh

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15 Slides • 15 Questions

Unit 6: Polynomial Functions

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About Unit 6: Polynomial Functions

One-day Lesson, One-day Review, Canvas Quiz
This Quizizz is the lesson and is graded for effort and completeness.
After we return from Spring Break, we will do a review and a practice quiz in Canvas.
Then we will move on to Unit 7: Rational Functions

What you need to be able to do with Polynomial Functions:

Determine the degree
Determine the zeros/roots/solutions
Determine the end behavior
Determine the increasing and decreasing intervals

Multiple Select

Which of these are polynomial functions?

$f\left(x\right)=5x+6$

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

$f\left(x\right)=\sqrt{x+8}$

$f\left(x\right)=\left|x-4\right|$

$f\left(x\right)=x^2+8x-9$

Polynomial Functions

No radicals
No absolute values
No variables in denominators

Multiple Select

Which of these are polynomial functions?

$f\left(x\right)=\frac{5x+6}{7}$

$f\left(x\right)=\frac{x^3-5x^2+x-9}{x+9}$

$f\left(x\right)=\sqrt[3]{x^2-3x}$

$f\left(x\right)=13$

$f\left(x\right)=2x-3$

Multiple Choice

What is the degree of the following polynomial? $f\left(x\right)=6x^3-5x^2+x+10$

Degree of a Polynomial

For polynomials with one variable, the degree is the highest exponent on the variable.

Multiple Choice

What is the degree of the following polynomial? $f\left(x\right)=7x^2+8x^5-x^3+14x-1+2x^4$

Multiple Choice

What are the approximate roots of the following function?

$-6,\ -1,\ 2.5$

$-4,\ 2.3$

$-6,\ -4,\ -1,\ 2.3,\ 2.5$

$2.3$

Zeros/Roots/Solutions

The x-values where the graph crosses or meets the x-axis.

For example, the roots of the graph shown are 1 and 4.

Multiple Choice

What are the approximate roots of the following function?

$-3,\ 0,\ 1,\ 2$

$-3,\ -1,\ 0,\ 0.5$

$-3,\ 1,\ 2$

$0,\ 1,\ 2$

Multiple Select

What is the end behavior of this graph? (Check all that apply.)

x→+∞,\ y→+∞

As $x→-∞,\ y→+∞$

As $x→+∞,\ y→-∞$

As $x→-∞,\ y→-∞$

End Behavior Symbols

As $x\longrightarrow-\infty$ "On the LEFT side..."
As $x\longrightarrow+\infty$ "On the RIGHT side..."
As $y\longrightarrow-\infty$ "The end points DOWN."
As $y\longrightarrow+\infty$ "The end points UP."

End Behavior in Words

"Where are the ends of the graph pointing?"
I. DOWN-DOWN
II. UP-UP
III. DOWN-UP
IV. UP-DOWN

End Behavior In Symbols

I. $As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty$ $As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty$
II. $As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty$ $As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty$
III. $As\ x\longrightarrow-\infty,\ y\longrightarrow-\infty$ $As\ x\longrightarrow+\infty,\ y\longrightarrow+\infty$
IV. $As\ x\longrightarrow-\infty,\ y\longrightarrow+\infty$ $As\ x\longrightarrow+\infty,\ y\longrightarrow-\infty$

Multiple Select

Which of the following describes an end behavior of the function graphed? Select all that apply.

As $x→-∞,\ y→-∞$

As $x→-∞,\ y→+∞$

As $x→+∞,\ y→-∞$

As $x→+∞,y→+∞$

Multiple Select

Which of the following describes an end behavior of the function graphed? Select all that apply.

As $x→-∞,\ y→-∞$

As $x→-∞,\ y→+∞$

As $x→+∞,\ y→-∞$

As $x→+∞,y→+∞$

Multiple Select

Which of the following describes an end behavior of the function graphed? Select all that apply.

As $x→-∞,\ y→-∞$

As $x→-∞,\ y→+∞$

As $x→+∞,\ y→-∞$

As $x→+∞,y→+∞$

Multiple Choice

When is this function increasing?

Interval: (-3, 3)

Interval: (-5, 5)

Interval: (-55, 55)

Interval: (0, 0)

Increasing and Decreasing Intervals

Remember the Roller Coaster?

This is review from Unit 1

Increasing Intervals

The roller coaster always travels from left to right.
"For what values of x is the roller coaster going uphill?"
From x=-4 to x=1, the roller coaster is going upwards.

Increasing Intervals

Answer is written as the interval $(-4,1)$ or $-4<x<1$

Multiple Select

When is this function increasing? (Select all that apply.)

Interval:

\left(-\infty,\ 1\right)

Interval:

\left(1,\ 3\right)

Interval:
$\left(3,\ 4\right)$

Interval:
$\left(4,+\infty\right)$

Increasing Intervals

Intervals $\left(1,3\right)\ and\ \left(4,\infty\right)$
or $1<x<3\ and\ 4<x<\infty$

Multiple Select

When is this function decreasing? (Check all that apply.)

Interval:

\left(-\infty,-1\right)

Interval: $\left(-\infty,1\right)$

Interval: $\left(-1,1\right)$

Interval: $\left(1,+\infty\right)$

Interval: $\left(-2,2\right)$

When is the function decreasing?

Intervals:

\left(-\infty,-1\right)\ and\ \left(1,+\infty\right)

Multiple Select

When is this function decreasing? (Check all that apply.)

Interval:

\left(-\infty,-4\right)

Interval: $\left(-4,-2\right)$

Interval: $\left(-2,\ 2\right)$

Interval: $\left(2,\ 5\right)$

Interval: $\left(5,\ +\infty\right)$

Next Class - Friday, March 12th

Friday is an Early Release Day. Classes are 50 minutes long and we get out at 11:15am.
Log into Meet, take attendance, listen to any announcements.
I will then stay on Meet to help anyone who has questions about any assignments from the third quarter.

Unit 6: Polynomial Functions

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