1. Solve by factoring: 2x 2 – 13x = Solve by quadratic formula: 8x 2 – 3x = Find the discriminant and fully describe the roots: 5x 2 – 3x. 4. Solve algebraically or graphically: x 2 – 2x – 15> 0 Algebra II 1

Graphing Polynomial Functions Algebra II

f(x) = a n x n + a n-1 x n a 1 x 1 + a 0 where a n ≠ 0 Example: f(x) = 3x 4 – 2x 3 + 5x – 4 Algebra II 3

 exponents are all ______________ therefore all __________________  all coefficients are___________________  a n is called the _____________________  a 0 is called the _____________________  n is equal to the ____________________ (always the _______________ exponent) Whole numbers Positive Real numbers Leading coefficient Constant term degree highest Algebra II 4

Standard Form means that the polynomial is written in _____________ order of _____________ Descending Exponents Algebra II 5

6

1. f(x) = ½ x 2 – 3x 4 – 7 2. f(x) = x x 3. f(x) = 6x 2 + 2x -1 + x 4. f(x) = -0.5x + πx 2 – √2 Yes f(x) = –3x 4 + ½x 2 – 7 D: 4 LC: -3 C: -7 N: Quartic Yes f(x) = πx x – √2 D: 2 LC: π C: –√2 N: Quadratic No exponents are not whole numbers No exponents are not whole numbers Algebra II 7

Direct Substitution means to: _____________________________________ ____ Plug the value into the equation and solve Algebra II 8

f(x) = 3x 3 – 2x 2 + 7x – 11 g(x) = – x 4 + 3x 2 + 2x + 7 p(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 = x 2 y = x 3 y = x 4 y = x 5 10 Algebra 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) = 3x 4 – 2x 2 + 5x – 8 D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 2. f(x) = -x 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) = x 7 – 3x 3 + 2x D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 4. f(x) = -2x 6 + 3x – 7 D: LC: End Behavior: f(x) --->____as x ----> -∞-∞ ∞ ∞ -∞-∞ ∞ -∞-∞-∞ -∞-∞ 1, positive 6, even -2, negative 7, odd Algebra II 14

5. f(x) = -4x 3 + 3x 8 D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 6. f(x) = 4x 3 + 5x 7 – 2 D: LC: End Behavior: 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) = – x x y Algebra II 17

18 Algebra II

2. f(x) = x 3 + x 2 – 4x – 1 x y Algebra II 19

20 Algebra II

3. f(x) = –x 4 – 2x 3 + 2x 2 + 4x x y Algebra II 21

22 Algebra II

4. f(x) = x 5 – 2 x y Algebra II 23

24 Algebra II

25 Algebra II Answer each: f(x) > 0 f(x) < 0 f(x) is increasing f(x) is decreasing

26 Algebra II Answer each: f(x) > 0 f(x) < 0 f(x) is increasing f(x) is decreasing

 f is increasing when x 4  f is decreasing when 0 < x < 4  f(x) >0 when -2 5  f(x) < 0 when x < -2 and 3 < x < 5 27 Algebra II Use the graph to describe the degree and the leading coefficient of f.

 f is decreasing when x 2.5  f is increasing when -1.5 < x < 2.5  f(x) >0 when x < -3 and 1 < x < 4  f(x) 4 28 Algebra 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 1  f(x) < 0 for all real numbers 29 Algebra 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) = t t t – Where t represents the year, with t = 1 corresponding to  Use a graphing calculator to graph the function for the interval 1 < t < 10. Describe the graph.  What was the average rate of change in the number of electric vehicles in use from 2001 to 2010? 30 Algebra II

The number of students S (in thousands) who graduate in four years from a university can be modeled by the function S(t) = -1/4t 3 + t , where t is the number of years since  Use a graphing calculator to graph the function for the interval 0 < t < 5. Describe the behavior of the graph on this interval.  What is the average rate of change in the number of four-year graduates from 2010 to 2015? 31 Algebra 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: 32 Algebra II 3. Give the end behavior for the function: 4. Graph: y = 2x 3 – 1