WARM-UP: 10/30/13Find the standard form of the quadratic function. Identify the vertex and graph.

23 5 1f x x x

WARM-UP: 10/26/12

23 5 1f x x x

21 3 5f x x x

2 51 3

3f x x x

2 51 3

25

36

25

12 3f x x x

2

13 53

12 6f x x

2

5 133

6 12f x x

5 13

,6 12

WARM-UP: 10/26/12

2

2 5 133 5 1 3

6 12f x x x f x x

1 2 3-1-2

1

2

-1

-2

-3

-4

-5

x

y

2.2 – POLYNOMIAL FUNCTIONS

IN THIS SECTION, YOU WILL LEARN TO

use transformation to sketch graphs

use the Leading Coefficient Test to determine the end behavior of polynomial graphs

use zeros of polynomial functions as sketching aids

DEFINITION OF A POLYNOMIAL FUNCTION:

1 2 21 2 2 1 0....

is a polynomial function of with degree .

n n nn n nf x a x a x a x a x a x a

x n

9 4 3Example: 4 8 3 2 7f x x x x x

PROPERTIES OF POLYNOMIAL FUNCTIONS:

1) Polynomial functions are continuous

Which of these is a polynomial function?

a) b)x

y

x

y

PROPERTIES OF POLYNOMIAL FUNCTIONS:

2) Polynomial functions only have smooth curves, not sharp turns.

Which of these is a polynomial function?

a) b)

x

y

x

y

LEADING COEFFICIENT TEST:

Definition of a leading coefficient: the coefficient of the highest degree

The Leading Coefficient Test determines the end-behavior of any polynomial function. It determines whether the graph falls or rises depending on the highest degree of the polynomial and its leading coefficient.

8 24 5 6f x x x

LEADING COEFFICIENT TEST:

The Leading Coefficient Test is dependent

on both of the following two values:

a) Highest degree of the polynomial: n can be odd or even

b) Leading coefficient:a can be positive or negative

LEADING COEFFICIENT TEST FOR ODD DEGREE POLYNOMIAL FUNCTIONS:

1) If a > 0 and n is odd, then the graph increases without bound on the right (rises) and decreases without bound on the left (falls).

3 23 2 1f x x x

Left End:

as

f x

x

Right End:

as

f x

x

1 2 3-1-2-3

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

LEADING COEFFICIENT TEST FOR ODD DEGREE POLYNOMIAL FUNCTIONS:

2) If a < 0 and n is odd, then the graph decreases without bound on the right (falls) and increases without bound on the left (rises).

3 23 2 1f x x x

Left End:

as

f x

x

Right End:

as

f x

x

1 2 3-1-2-3

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

LEADING COEFFICIENT TEST FOR EVEN DEGREE POLYNOMIAL FUNCTIONS:

1) If a > 0 and n is even, then the graph increases without bound on the right and on the left (rises).

4 23 4 1f x x x

Left End :

as

f x

x

Right End:

as

f x

x

1 2 3-1-2-3

1

2

3

4

5

6

-1

-2

x

y

LEADING COEFFICIENT TEST FOR EVEN DEGREE POLYNOMIAL FUNCTIONS:

2) If a < 0 and n is even, then the graph decreases without bound on the right and on the left (falls).

4 23 4 1f x x x

Left End:

as

f x

x

Right End:

as

f x

x

1 2 3-1-2-3

1

2

3

-1

-2

-3

-4

-5

x

y

LEADING COEFFICIENT TEST SUMMARY:

Degree n

Sign of a

Left-End

Right-End

odd a > 0 Falls Rises

odd a < 0 Rises Falls

even a > 0 Rises Rises

even a < 0 Falls Falls

LEADING COEFFICIENT TEST:

Note: The Leading Coefficient Test only determines the end behavior of the function, but does not tell you how the graph behaves between the end behavior.

ZEROS OF THE FUNCTION:

The zeros of the function can help determine

certain properties of the polynomial graph.

a) The function can have at most n -1 turns: points at which the graph can change from increasing to decreasing.

ZEROS OF THE FUNCTION:

2) The function can have at most n real zeros.

Since the highest degree is 6, this function

can have most 6 real zeros.

64 3 3f x x x

ZEROS OF THE FUNCTION:

3) In general, multiple roots will behave in two different ways.

a) If k is even, the graph will only touch the x-axis and not cross it.

b) If k is odd, the graph will cross the x-axis.

kx a

2 33 5f x x x

2 313 2

4f x x x

2a) 3 : only touches the axis at 3 x x x

3b) 2 : will cross the axis at 2x x x

1 2 3 4 5-1-2-3-4-5

5

10

15

20

25

30

35

-5

-10

-15

-20

x

y

2 313 2

4f x x x

2a) 3 : only touches the axis at 3 x x x

3b) 2 : will cross the axis at 2x x x

1 2 3 4 5-1-2-3-4-5

5

10

15

20

25

30

-5

-10

-15

-20

x

y

3 222 3 2 1f x x x x x

22a) 2 , 1 : only touches the axis at 1,0 x x x x

3b) 3 , 2 : will cross the axis at 2,3x x x x

1 2 3 4 5-1-2-3-4

50

100

150

200

250

300

350

-50

-100

-150

x

y

3 222 3 2 1f x x x x x

22a) 2 , 1 : only touches the axis at 1,0 x x x x

3b) 3 , 2 : will cross the axis at 2,3x x x x

1 2 3 4 5-1-2-3-4

50

100

150

200

250

300

350

-50

-100

-150

x

y

STEPS TO GRAPH A FUNCTION:

1) Solve for the zeros2) Solve for the y-intercept3) Use the information from the multiple

roots to determine where it touches and crosses

4) Use the Leading Coefficient Test to determine the end behavior

5) Plot a few points between the zeros

GRAPH THE FUNCTION:

3 213 2

4f x x x

3c) 3 : will cross the axis at 3 x x x

2d) 2 : will touch the axis at 2 x x x

e) Since 5 and 0, the graph will rise

on the left and fall on the right 

n a

a) Zeros : 2,3 x

b) intercept : 27 y y

GRAPH THE FUNCTION:

3 213 2

4f x x x

1 2 3 4 5-1-2-3-4-5

5

10

15

20

25

30

-5

-10

x

y

GRAPH THE FUNCTION:

3 213 2

4f x x x

1 2 3 4 5-1-2-3-4-5

5

10

15

20

25

30

-5

-10

x

y

GRAPH THE FUNCTION:

c) 3 : will cross the axis at 3 x x x

d) 2 : will cross the axis at 2 x x x

f) Since 4 and 0, the graph will fall

on the left and fall on the right 

n a

a) Zeros : 2,0,3 xb) intercept : 0 y y

2 23 6f x x x x

2e) 3 : will touch the axis at 0 x x x

23 3 2x x x 4 3 23 3 18f x x x x

GRAPH THE FUNCTION:

4 3 2 23 3 18 3 3 2f x x x x x x x

1 2 3 4 5-1-2-3-4-5

10

20

30

40

50

-10

-20

x

y

GRAPH THE FUNCTION:

4 3 2 23 3 18 3 3 2f x x x x x x x

1 2 3 4 5-1-2-3-4-5

10

20

30

40

50

-10

-20

x

y

GRAPH THE FUNCTION:

2c) 1 : will touch the axis at 1 x x x

f) Since 5 and 0, the graph will fall

on the left and rise on the right 

n a

a) Zeros : 0,1 xb) intercept : 0 y y

3 23 2 1f x x x x

3d) 3 : will cross the axis at 0 x x x

233 1x x

5 4 33 6 3f x x x x

GRAPH THE FUNCTION:

25 4 3 33 6 3 3 1f x x x x f x x x

1 2 3 4 5-1-2-3-4-5

4

8

12

16

20

-4

-8

-12

-16

-20

x

y

GRAPH THE FUNCTION:

25 4 3 33 6 3 3 1f x x x x f x x x

1 2-1-2

1

-1

-2

x

y

GRAPH THE FUNCTION:

Since this is not factorable, we have to use the rational root theorem and synthetic division to solve for the zeros of the function.

3 22 7 2f x x x x

GRAPH THE FUNCTION:

Rational Root Theorem:

p: the factors of the constantq: the factors of the leading coefficient

3 22 7 2f x x x x

2, 1 1

p

q

2, 1Therefore, 2, 1

1

p

q

GRAPH THE FUNCTION:

Synthetic Division:

3 22 7 2f x x x x

1 1 2 7 2 1 1 2 7 2

______________ _______________

0 0

2 1 2 7 2 2 1 2 7 2

______________ _______________

0 0

GRAPH THE FUNCTION:

Synthetic Division:

We now have to use the quadratic equation to solve for the remaining zeros.

3 22 7 2f x x x x

2

2 1 2 7 2

2 8 2

1 4 1 0 4 1 0x x

2 4

2

b b acx

a

Depressed Equation

GRAPH THE FUNCTION:

Quadratic Equation:

Zeros of the function: y-intercept:

2 4 1f x x x

2 4 1

4 16 4 4 12 4 2 32 2 3

2 2 20.26,3.73

f x x x

x

x

2,0.26,3.73x2y

GRAPH THE FUNCTION:

3 22 7 2f x x x x

Leading Coefficient Test:

Right-End: as

Left-End: as

f x x

f x x

GRAPH THE FUNCTION:

3 22 7 2f x x x x

1 2 3 4 5-1-2-3-4-5

2

4

6

8

-2

-4

-6

-8

-10

-12

-14

x

y