1. Warm-up:
Find an equation of the parabola in standard form given the vertex at (2, 3) and that it goes through the point (4, -1).
HW: page 227(9-11, 13-22)

2. G 10. a. b. C B H c. d. F A E D 11. a. b. a. b. c. d. c. d.
HW Answers: page 214 (1-12, , , 47-50)
G a. b. C B
H c d. F A E
D a b.
a b.
c d c d.

3. HW Answers: page 214 (1-12, 27- 30, 37- 40, 47-50)

4. 37) f(x) = (x + 2)2 + 5 49) (-5/2, 0), (6, 0) 38) f(x) = (x – 4)2 - 1
HW Answers: page 214 (1-12, , , 47-50)
37) f(x) = (x + 2) ) (-5/2, 0), (6, 0)
38) f(x) = (x – 4)2 - 1
39) f(x) = -1/2(x – 3)2 + 4
40) f(x) = -1/4(x – 2)2 + 3
47) (0, 0), (4, 0)
50) (7, 0), (-1, 0)
48) (3, 0), (6, 0)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

5. Polynomial Functions of a Higher Degree
Objective:
Use transformations to sketch graphs of polynomial functions
Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions

6. Polynomial Function Leading Coefficient Degree
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

7. Polynomial functions are continuous no breaks, holes, or gaps.x y
f (x) = x3 – 5x2 + 4x + 4 x y x y
continuous
not continuous
continuous
smooth
piecewise
not smooth
polynomial
not polynomial
not polynomial
Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

8. Polynomial functions of the form f (x) = x n, n  1 are called power functions.
If n is even, their graphs resemble the graph of f (x) = x2.
If n is odd, their graphs resemble the graph of f (x) = x3.

9. Example: Sketch the graph of f (x) = x4 – 3
This is a shift of the graph of f(x) = x 4 three units down. x y

10. Example: Sketch the graph of f (x) = – x3 .
This is a reflection of the graph of f(x) = x 3 in the x-axis. x y
f (x) = -x3
f (x) = x3

11. Example: Sketch the graph of f (x) = – (x + 2)4 .
This is a shift of the graph of y = – x 4 two units to the left and a reflection of the graph of y = x 4 in the x-axis. x y
y = x4
f (x) = – (x + 2)4
y = – x4

12. Example: Sketch the graph of f (x) = – (x + 2)3 – 3 .
This is a reflection of the graph of f(x) = x 3 in the x-axis. x y
f (x) = -x3
f (x) = x3
f (x) = -(x + 2)3
f (x) = -(x + 2)3 – 3

13. f(x) = – (x + 2)2 + 3 Shape Reflect about x-axis Shift up 3y
Shift left 2
f (x) = x2
f (x) = – (x + 2)2 + 3

14. Left and right opposite
Leading Coefficient Test
As x grows positively or negatively without bound, the value f (x) of the polynomial function
f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0)
grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n is odd or even. x y x y
an positive
Rises right
an negative
Falls right
n even
Left and right same
n odd
Left and right opposite

15. Applying the leading coefficient test
Use the leading coefficient test
to describe left and right hand behavior of the graph
Example 1 y x –2 2
Oddopposite on left (rises left)
(-) Falls to the right

16. Applying the leading coefficient test
Use the leading coefficient test
to describe left and right hand behavior of the graph y x –2 2
Example 2
EvenSame on left (rises left)
(+) Rises to the right

17. Example: Describe the right-hand and left-hand behaviour for the graph of f(x) = –2x3 + 5x2 – x + 1.
Negative -2
Leading Coefficient
Odd 3
Degree x y
Rises to the left
Falls to the right

18. Polynomial Functions of a Higher Degree
Summary:
Use transformations to sketch graphs of polynomial functions
Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions

19. Sneedlegrit: HW: page 227(9-11, 13-22)
Use the leading coefficient test
to describe left and right hand behavior of the graph y x –2 2
HW: page 227(9-11, 13-22)