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)

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, 27- 30, 37- 40, 47-50) 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, 27- 30, 37- 40, 47-50)    

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, 27- 30, 37- 40, 47-50) 37) f(x) = (x + 2)2 + 5 49) (-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.

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

Polynomial Function Leading Coefficient Degree Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree -2 5 1 3 14 0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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.

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.

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

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

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

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

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

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

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

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

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

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

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)