Warm Up Sept. 21 Sit 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. x x y 2 – 9 y – x 2 – x 3 – x x – 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 a 0, a 1, a 2, …, a n-1, a n be real numbers with a n ≠ 0. The function given by f(x) = a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x + a 0 is a polynomial function of degree n. The leading coefficient is a n.

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) = ax 2 + bx + c

Some examples of polynomial functions: Cubic Function: f(x) = ax 3 + bx 2 + cx + d

Some examples of polynomial functions: Quartic Function: f(x) = ax 4 + 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) = 3x 4 – 5x 2 – 1 2.f(x) = -3x 2 – 2x 7 – 4x 4 3.f(x) = x 3 – 2x 2

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. a n positivea n 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. a n positivea n negative

Let’s describe the end behavior of the following polynomials: 1.f(x) = 3x 4 + 5x 5 + 2x 3 2.f(x) = 21 – x 4 – 4x 2 3.f(x) = x 6 – 8x x 4

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) = 3x 4 + 5x 5 + 2x 3 2.f(x) = 21 – x 4 – 4x 2 3.f(x) = x 6 – 8x x 4

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) = 3x 4 + x 5 + 2x 3 2.f(x) = 21 – x 4 – 4x 2 3.f(x) = x 6 – 8x x 4

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them. f(x) = 3x 4 + 5x 5 + 2x 3

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them. f(x) = 21 – x 4 – 4x 2

So now…for those 3 functions, we know their end behavior, number of turning points, and zeroes. Let’s graph them. f(x) = x 6 – 8x x 4

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) = x 4 – 4x 3 – 32x 2

How many turning points and zeroes can it have? f(x) = x 4 – 4x 3 – 32x 2

Find the zeroes by factoring. f(x) = x 4 – 4x 3 – 32x 2

Graph the function. f(x) = x 4 – 4x 3 – 32x 2

Write the end behavior using limits f(x) = -9x x 4

How many turning points and zeroes can it have? f(x) = -9x x 4

Find the zeroes by factoring. f(x) = -9x x 4

Graph the function. f(x) = -9x x 4

Write the end behavior using limits F(x) = 6x 5 – 150x 3

How many turning points and zeroes can it have? F(x) = 6x 5 – 150x 3

Find the zeroes by factoring. F(x) = 6x 5 – 150x 3

Graph the function. F(x) = 6x 5 – 150x 3

Write the end behavior using limits F(x) = -11x 4 – 3x x 3

How many turning points and zeroes can it have? F(x) = -11x 4 – 3x x 3

Find the zeroes by factoring. F(x) = -11x 4 – 3x x 3

Graph the function. F(x) = -11x 4 – 3x x 3

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) = x 5 + 3x 4 – 10x 3 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.