1. Solve by factoring: 2x2 – 13x = 15.

2. Solve by quadratic formula: 8x2 – 3x = 10.

3. Find the discriminant and fully describe the roots: 5x2 – 3x.

4. Solve algebraically or graphically: x2 – 2x – 15> 0

Algebra II 1

Graphing Polynomial Functions

Algebra II

f(x) = anxn + an-1xn-1 + ... + a1x1 + a0

where an ≠ 0

Example: f(x) = 3x4 – 2x3 + 5x – 4

Algebra II 3

exponents are all ______________ therefore all __________________

all coefficients are___________________

an is called the _____________________

a0 is called the _____________________

n is equal to the ____________________ (always the _______________ exponent)

Whole numbersPositive

Real numbers

Leading coefficient

Constant term

degreehighest

Algebra II 4

Standard Form means that the polynomial is written in _____________ order of _____________

Descending Exponents

Algebra II 5

Standard Form Example Degree Name

f(x) = a0

f(x) = a1x1 + a0

f(x) = a2x2 + a1x1 + a0

f(x) = a3x3 + a2x2 + a1x1 + a0

f(x) = a4x4 + a3x3 + a2x2 + a1x1 + a0

Algebra II 6

1. f(x) = ½ x2 – 3x4 – 7

2. f(x) = x3 + 3x

3. f(x) = 6x2 + 2x-1 + x

4. f(x) = -0.5x + πx2 – √2

Yesf(x) = –3x4 + ½x2 – 7

D: 4LC: -3C: -7

N: Quartic

Yesf(x) = πx2 - 0.5x – √2

D: 2LC: πC: –√2

N: Quadratic

Noexponents are not

whole numbers

Noexponents are not

whole numbers

Algebra II 7

Direct Substitution means to:

_________________________________________Plug the value into the equation and solve

Algebra II 8

f(x) = 3x3 – 2x2 + 7x – 11g(x) = – x4 + 3x2 + 2x + 7p(x) = – x(2x – 3)(x + 7)

1. p(2) 2. g(3) 3. f(-2) 4. g(-3)–18 –

41–57 –53

Algebra II 9

Lets type each in the calculator and look for:

y = x y = x2

y = x3

y = x4 y = x5

10Algebra II

End behavior is what the y values are doing as the x values approach positive

and negative infinity.

It is written: f(x) _____ as x -∞, and

f(x) _____ as x ∞

Algebra II 11

If the degree is __________ the ends of the graph go in the _________ direction.

If the degree is __________ the ends of the graph go in the _________ directions.

Look at the ________________ to see what direction the graph is going in.

odd

same

opposite

Leading coefficient

even

Algebra II 12

1. f(x) = 3x4 – 2x2 + 5x – 8

D:

LC:

End Behavior:

f(x) --->____ as x --->

f(x) --->____ as x ---->

2. f(x) = -x2 + 1

D:

LC:

End Behavior:

f(x) --->____as x --->

f(x) --->____ as x ---->

-∞

∞∞

-∞

∞

∞ -∞

-∞

3, positive

2, even

-1, negative

4, even

Algebra II 13

3. f(x) = x7 – 3x3 + 2x

D:

LC:

End Behavior:

f(x) --->____ as x --->

f(x) --->____ as x ---->

4. f(x) = -2x6 + 3x – 7

D:

LC:

End Behavior:

f(x) --->____as x ---->

f(x) --->____ as x ---->

-∞

∞∞

-∞

∞

-∞ -∞

-∞

1, positive

6, even

-2, negative

7, odd

Algebra II 14

5. f(x) = -4x3 + 3x8

D:

LC:

End Behavior:

f(x) --->____ as x --->

f(x) --->____ as x ---->

6. f(x) = 4x3 + 5x7 – 2

D:

LC:

End Behavior:

f(x) --->____as x ---->

f(x) --->____ as x ---->

-∞

∞∞

-∞

∞

∞ -∞

∞

3, positive

7, odd

5, positive

8, even

Algebra II 15

1. Make a table of values from -3 to 3

2. Plot the points

3. Connect with a smooth curve**(use arrows to demonstrate end behavior)**

Algebra II 16

1. f(x) = – x3 + 1

x y -3-2-10123

289210-7-26

Algebra II 17

18Algebra II

2. f(x) = x3 + x2 – 4x – 1

x y -3-2-10123

-733-1-3323

Algebra II 19

20Algebra II

3. f(x) = –x4 – 2x3 + 2x2 + 4x

x y -3-2-10123

-210-103-16-105

Algebra II 21

22Algebra II

4. f(x) = x5 – 2

x y -3-2-10123

-245-34-3-2-130241

Algebra II 23

24Algebra II

25Algebra II

Answer each: f(x) > 0

f(x) < 0

f(x) is increasing

f(x) is decreasing

26Algebra II

Answer each: f(x) > 0

f(x) < 0

f(x) is increasing

f(x) is decreasing

f is increasing when x < 0 and x > 4 f is decreasing when 0 < x < 4 f(x) >0 when -2 < x

< 3 and x >5 f(x) < 0 when x < -2

and 3 < x < 5

27Algebra II

Use the graph to describe the degree and the leading coefficient of f.

f is decreasing when x < -1.5 and x > 2.5 f is increasing when -1.5 < x < 2.5 f(x) >0 when x < -3

and 1 < x < 4 f(x) < 0 when -3 < x <

1 and x > 4

28Algebra II

Use the graph to describe the degree and the leading coefficient of f.

f is increasing when x < -1 and 0 < x < 1 f is decreasing when -1 < x < 0 and x > 1 f(x) < 0 for all real

numbers

29Algebra II

Use the graph to describe the degree and the leading coefficient of f.

The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function v(t) = .151280t3 - 3.28234t2 + 23.7565t – 2.041Where t represents the year, with t = 1 corresponding to 2001. a.Use a graphing calculator to graph the function for the interval 1 < t < 10. Describe the graph. b.What was the average rate of change in the number of electric vehicles in use from 2001 to 2010?

30Algebra II

The number of students S (in thousands) who graduate in four years from a university can be modeled by the function S(t) = -1/4t3 + t2 + 23, where t is the number of years since 2010. a.Use a graphing calculator to graph the function for the interval 0 < t < 5. Describe the behavior of the graph on this interval. b.What is the average rate of change in the number of four-year graduates from 2010 to 2015?

31Algebra II

1. Decide whether the function is a polynomial function. If it is, write the function in standard from and state the degree and leading coefficient:

2. Use direct substitution to find f(-1) for the function:

32Algebra II

3. Give the end behavior for the function:

4. Graph: y = 2x3 – 1