Warm Up Sept. 21Sit in your usual seat with your tracking sheet and

homework on your desk. If you didn’t get a textbook yesterday and would like one, let me know now.

FACTOR THE FOLLOWING:

1. x2 + 5x + 6

2. 2y2 – 9y – 5

3. 9x2 – 64

4. -15x3 – x2 + 15x – 3

Intro to Polynomial Functions

Unit 3 Objectives• Use power functions to model and solve

problems; justify results.• Solve using tables, graphs, and algebraic

properties.• Interpret the constants, coefficients, and bases in

the context of the problem.• Create and use calculator-generated models

of polynomial functions of bivariate data to solve problems.• Check models for goodness-of-fit; use the most

appropriate model to draw conclusions and make predictions.

Today’s Objectives• SWBAT recognize and graph polynomial

functions.• Definition of polynomial functions• What are the components?• Some examples• Even vs. Odd

• SWBAT determine the end behavior of a function by looking at its leading coefficient.

• SWBAT find the zeroes and number of possible turning points of a function and use this information to graph it!

What is a polynomial function?Let n be a nonnegative integer and let a0,

a1, a2, …, an-1, an be real numbers with an ≠ 0. The function given by

f(x) = anxn + an-1 xn-1 + … + a2 x2 + a1 x + a0

is a polynomial function of degree n. The leading coefficient is an.

In other words…

• The SUMS and DIFFERENCES of monomials form other types of polynomials.

Some examples of polynomial functions:• Constant Function f(x) = c

Some examples of polynomial functions:• Linear Function: f(x) = ax + c

Some examples of polynomial functions:• Quadratic Function: f(x) = ax2 + bx + c

Some examples of polynomial functions:

• Cubic Function: f(x) = ax3 + bx2 + cx + d

Some examples of polynomial functions:

• Quartic Function: f(x) = ax4 + bx3 + cx2 + dx + e

You get the idea.

• Let’s learn how to graph them.• In order to graph them, we need to

know:• End Behavior• Turning Points• Zeroes

The important components of a polynomial function:• Coefficients

• Constant

• Exponents

The most important coefficient is the….• LEADING COEFFICIENT

• The leading coefficient is the number in front of the term with the largest exponent.

• It is important because it tells us about the end behavior of a function.• (We’ll get there in a minute)

What are the leading coefficients of the following polynomials?1. f(x) = 3x4 – 5x2 – 1

2. f(x) = -3x2 – 2x7 – 4x4

3. f(x) = x3 – 2x2

End Behavior

• As functions go on forever, they can go one of two ways:

• Up towards • Or down towards –

• The leading coefficient tells us what they do!

Even Degree Functions (the largest exponent is even)• They all sort of look like quadratic

functions.• That means their ends are either BOTH

going up, or BOTH going down.an positive an negative

Odd Degree Functions (the largest exponent is odd)• They all sort of look like cubic functions.• That means that one end goes up, and

the other one goes down.

an positive an negative

Let’s describe the end behavior of the following polynomials:1. f(x) = 3x4 + 5x5 + 2x3

2. f(x) = 21 – x4 – 4x2

3. f(x) = x6 – 8x5 + 12x4

Turning Points

• These are the maximums and minimums we learned to identify last week.

• They are where the graph CHANGES DIRECTION.

• A polynomial function has one less turning point than the value of its largest exponent.

How many turning points could these functions have?.1. f(x) = 3x4 + 5x5 + 2x3

2. f(x) = 21 – x4 – 4x2

3. f(x) = x6 – 8x5 + 12x4

Zeroes• Zeroes are the places where the graph

crosses the x axis.• f(x) = 0 at these points.• A graph CAN have as many zeroes as

the value of its biggest exponent. (But it doesn’t have to.)

• We find zeroes by factoring the polynomial and then setting each factor equal to zero and solving for x.

Let’s find the zeroes of the following functions.1. f(x) = 3x4 + x5 + 2x3

2. f(x) = 21 – x4 – 4x2

3. f(x) = x6 – 8x5 + 12x4

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them.• f(x) = 3x4 + 5x5 + 2x3

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them.• f(x) = 21 – x4 – 4x2

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them.• f(x) = x6 – 8x5 + 12x4

Whiteboards!

• Let’s practice determining end behavior, finding zeroes, and graphing.

• When I put a problem on the board, complete that problem on your white board, and hold it up when I ask for it.

• Ready?!?!

Write the end behavior using limits• f(x) = x4 – 4x3 – 32x2

How many turning points and zeroes can it have?• f(x) = x4 – 4x3 – 32x2

Find the zeroes by factoring.

• f(x) = x4 – 4x3 – 32x2

Graph the function.

• f(x) = x4 – 4x3 – 32x2

Write the end behavior using limits• f(x) = -9x6 + 36x4

How many turning points and zeroes can it have?• f(x) = -9x6 + 36x4

Find the zeroes by factoring.

• f(x) = -9x6 + 36x4

Graph the function.

• f(x) = -9x6 + 36x4

Write the end behavior using limits• F(x) = 6x5 – 150x3

How many turning points and zeroes can it have?• F(x) = 6x5 – 150x3

Find the zeroes by factoring.

• F(x) = 6x5 – 150x3

Graph the function.

• F(x) = 6x5 – 150x3

Write the end behavior using limits• F(x) = -11x4 – 3x5 + 20x3

How many turning points and zeroes can it have?• F(x) = -11x4 – 3x5 + 20x3

Find the zeroes by factoring.

• F(x) = -11x4 – 3x5 + 20x3

Graph the function.

• F(x) = -11x4 – 3x5 + 20x3

Write the end behavior using limits

How many turning points and zeroes can it have?

Find the zeroes by factoring.

Graph the function.

Write the end behavior using limits

How many turning points and zeroes can it have?

Find the zeroes by factoring.

Graph the function.

Exit Ticket

• Given the following function: f(x) = x5 + 3x4 – 10x3

a. Write its end behavior using limits.b. How many possible turning points does it have?c. How many possible zeroes does it have? d. Find the zeroes.e. Sketch a graph of the function!

Homework

• Factor puzzle.• This homework will count for your

homework stamp on Monday as well as a class work grade.