1. AFM Objective 1.01

2.  Polynomials have more than one term: ◦ 2x + 1 ◦ 3x 2 + 2x + 1  The fancy explanation: ◦ f(x) = a n x n + a n-1 x n-1 + …+ a 2 x 2 + a 1 x + a ◦ Ex: 4x 3 + 3x 2 + 2x + 1  Simply put: the addition and subtraction of terms with different exponents going from largest to smallest

3.  Polynomial functions are named by their degrees  The largest exponent is the degree of the function

4.  Leading Term: ◦ The term with the highest exponent ◦ This function has a degree of 3 ◦ By knowing the leading term we can analyze a function without graphing it *The highest exponent may not be written first in the function!!

5.  What are the degrees of the following polynomial functions? 1. 3x 5 – 2x 4 + 9x – 2 2. -6x 3 + 2x 2 – 7x + 10 3. 2x 11 + 5x 6 – 3x 2 4. 4x 2 + 5x 4 – 2x 3 + x – 5

6.  What are the degrees of the following polynomial functions? 1. 3x 5 – 2x 4 + 9x – 2 2. -6x 3 + 2x 2 – 7x + 10 3. 2x 11 + 5x 6 – 3x 2 4. 4x 2 + 5x 4 – 2x 3 + x – 5

7.  Certain polynomial functions have similar qualities.  We will use the Leading Term to analyze the behavior of a function  For simplicity sake we will just use the function f(x) = ax n

8.  If a is positive and n is an odd number what kind of equations will you get? ◦ 2x ◦ 4x 3 ◦ 7x 5  Here’s an example:

9.  Here’s another example:  Notice how the straight line and this line both point down on the left and up on the right  The have the same “end behavior”

10.  All equations that have a positive a and odd n have the same end behavior

11.  If a is negative and n is odd it will change the end behavior (ex: -2x 3 )

12.  If a is positive and n is an even number what kind of equations will you get? ◦ 2x 2 ◦ 4x 4 ◦ 7x 6  Here’s an example:  Notice that both ends point the same way, up!

13.  All equations where a is positive and n is an even number will have the same end characteristics:

14.  If a is negative and n is an even number it will change the end behavior  Here’s an example:  Notice that both ends still point the same way but down

15.  All equations where a is negative and n is an even number the end behavior will be:

16.  What is the end behavior of the following functions? 1. f(x) = 3x 3 2. f(x) = 4x 6 – 2x 2 + 1 3. f(x) = -3x 2 + 2x – 5 4. f(x) = 4x – 5x 5 + 3x 3

17.  What is the end behavior of the following functions? 1. f(x) = 3x 3 2. f(x) = 4x 6 – 2x 2 + 1 3. f(x) = -3x 2 + 2x – 5 4. f(x) = 4x – 5x 5 + 3x 3

18.  Turning Points: ◦ Where the graph changes direction ◦ It goes from increasing to decreasing or the other way around

19.  This is a cubic (x 3 ) function and it has 2 turning points

20.  This is also a cubic (x 3 ) function  Notice there are no Turning Points!  This function never changes it’s direction *Cubic (x 3 ) functions can have up to 2 Turning Points

21.  This is a quartic (x 4 ) function

22.  This is also a quartic (x 4 ) function  Notice there is just one Turning Point!  This function only changes direction once *Quartic (x 4 ) functions can have up to 3 Turning Points

23.  If you look at the exponent on the Leading Term you can analyze up to how many Turning Points there can be

24.  Up to how many possible turning points do the following functions have? 1. 3x 5 – 2x 4 + 9x – 2 2. -6x 3 + 2x 2 – 7x + 10 3. 2x 11 + 5x 6 – 3x 2 4. 4x 2 + 5x 4 – 2x 3 + x – 5

25.  Up to how many possible turning points do the following functions have? 1. 3x 5 – 2x 4 + 9x – 2 2. -6x 3 + 2x 2 – 7x + 10 3. 2x 11 + 5x 6 – 3x 2 4. 4x 2 + 5x 4 – 2x 3 + x – 5

26.  A zero is the x value when the function crosses the x-axis  This is a quartic (x 4 ) function  Notice it has 4 zeros

27.  This is also a quartic (x 4 ) function  Notice it has only 2 zeros  The others are “imaginary” *Quartic functions can have up to 4 real zeros

28.  If you look at the exponent on the Leading Term you can analyze up to how many Zeros there can be

29.  Remember that when a line crosses the x- axis, the y value is zero.  Any point that is (x, 0) is a “zero” or x- intercept  You can find some of the zeros by solving the equations

30.  Example: Determine all the real zeros for  This is a cubic function so it can have up to 3 real zeros  First set the function equal to zero (y is zero)  We need to factor to find the solutions

31.  Find a common factor and pull it out  Factor what’s left inside the parenthesis

32.  Set each part of the function equal to zero  Solve each one for x

33.  Example 2: Determine all the real zeros for  We know this will have up to 4 real zeros  Also, this equation is in quadratic form since we can write it like:

34.  In order to factor this properly we can replace the x 2 ’s with the letter u  We now have an equation we can factor!

35.  Factor u 2 – 8u + 15  Remember, u is equal to x 2  Plug x 2 in for u

36.  Solve each one for x

37.  Determine all the real zeros for  This can have up to 4 real zeros  Factor out the GCF first (always take the negative with the first number)

38.  Factor inside the parenthesis  Set everything equal to zero

39.  Solve each part for x

40.  Notice that you have two answers of zero  This is called “Multiplicity”  Multiplicity- when you have a repeated zero (answer)

41.  Solve each part for x