WarmUp Please fill out the chart to the

Warm-Up �Please fill out the chart to the best of your ability

Assignment �p. 168 �# 1, 4, 6

Section 1. 5 Shifting, Reflecting, and Stretching Graphs Objectives: Students will know how to identify and graph shifts, reflections, and nonrigid transformations of functions. Extra Practice http: //www. khanacademy. org/exercise/shifting_and_reflecting_functions Extra Examples http: //www. khanacademy. org/video/algebra-ii--shifting-quadratic-graphs? topic=californiastandards-test-algebra-2

The Original Six �Constant Function f(x) = c X Y -1 -2 0 -2 1 -2 �Identity Function f(x) = x X Y -1 -1 0 0 1 1

Absolute Value Function f(x) = |x| X Y -1 1 0 0 1 1 Square Root Function X Y -1 0 0 4 2

Quadratic Function f(x) = x 2 X Y -2 4 0 0 2 4 Cubic Function f(x) = x 3 X Y -1 -1 0 0 2 8

Vertical and Horizontal Shifts �Use a graphing utility to graph: � Y 1 = f(x) = x 2. Then, on the same viewing screen, graph Y 2 = (x – 4)2. � How did we change the equation? � How did the graph change? �Y 3 = (x + 4)2, Y 4 = x 2 – 4, and Y 5 = x 2 + 4. � How did we change the equations? � How did the graphs change?

�Let c be a positive real number. The following changes in the function y = f(x) will produce the stated shifts in the graph of y = f(x). �h(x) =f(x - c) �Y 2 = (x – 4)2 �h(x) =f(x + c) �Y 3 = (x + 4)2 �h(x) =f(x) - c �Y 4 = x 2 – 4 �h(x) =f(x) + c �Y 5 = x 2 + 4 Horizontal shift c units to the right Horizontal shift c units to the left Vertical shift c units downward Vertical shift c units upward

Example 1. Given f(x) = x 3 + x, describe and graph the shifts in the graph of f generated by the following functions. a) g(x) = (x + 1)3 + x + 1. b) h(x) = (x - 4)3 + x.

�Let. Write the equation for the function resulting from a vertical shift of 3 units downward and a horizontal shift of 2 units to the right of the graph of f(x) = | x 2 | 3

Warm Up �Write about what it means to reflect over the y-axis and x-axis without using the word symmetry?

Assignment �http: //www. khanacademy. org/exercise/shifting_and_ reflecting_functions �Register me as your coach and do 10 problems

Reflecting Graphs

Reflecting Graphs �Use a graphing utility to graph: � Y 1 = f(x) = (x – 2)3. Then, on the same viewing screen, graph Y 2 = -(x – 2)3. �Y 3 = (-x - 2)3.

�The following changes in the function y = f(x) will produce the stated reflections in the graph of y = f(x). �h(x) =f(-x) Reflection in the y-axis �h(x) = -f(x) Reflection in the x-axis

�Example 2. Given f(x) = x 3 + 3, describe the reflections in the graph of f generated by the following functions. � a) g(x) = -x 3 + 3. Reflected in the ? ? ? -axis. � b) h(x) = -(x 3 + 3) = -x 3 - 3. Reflected in the ? ? ? -axis.

�Example 3. Below is the graph of �a) y = b) Graph y = -f(x). c) Graph y = f(-x) + 1 1. 2. 3.

Widening and Narrowing �Distort the shape of the graph �Is not shifting or reflecting it. �Come from equations of the form y = cf(x). �If c > 1, then there is a vertical stretch of the graph of y = f(x). If 0 < c < 1, then there is a vertical shrink.

Example 4. Given f(x) = 1 - x 2, describe the graph of g(x) = 3 – 3 x 2. �Because 3 – 3 x 2 = 3(1 - x 2), the graph of g is a vertical stretch (each y-value is multiplied by 3) of the graph of f. X f(x)=1 - x 2 g(x) = 3 – 3 x 2 -1 0 0 0 1 3 1 0 0 2 -3 -9

X Y X Y -2 4 -2 8 -2 4 -2 (8/3) -1 1 -1 2 -1 1 -1 (2/3) 0 0 0 0 1 1 1 2 1 1 1 (2/3) 2 4 2 8 2 4 2 (8/3)

�Please describe the following function �g(x) = -2 f(x) �Reflection? Narrower? Wider or �h(x) = �Reflection? Wider or Narrower

Warm Up In the mail, you receive a coupon for $5 off of a pair of jeans. When you arrive at the store, you find that all jeans are 25% off. You find a pair of jeans for $36. � 1. If you use the $5 off coupon first, and then you use the 25% off on the remaining amount, how much will the jeans cost? � 2. If you use the 25% off first, and then you use the $5 off on the remaining amount, how much will the jeans cost?

Jean fiend �Let the cost of the jeans be represented by a variable x. Write a function f(x) that represents the cost of the jeans after the $5 off coupon. �Write a function g(x) that represents the cost of the jeans after the 25% discount.

Function Composition �Write a new function r(x) that represents the cost of the jeans if the 25% discount is applied first and the $5 off coupon is applied second. �Write a new function s(x) that represents the cost of the jeans if the $5 off coupon is applied first and the 25% discount is applied second.

Compositions of Functions �The composition of the function f with the function g is � (f �g)(x) = f(g(x)). �f(x) = x – 5, g(x) =. 75 x �(f �g)(x) = f(g(x)) = [. 75 x] - 5 �The composition of the function g with the function f is � (g �f )(x) = g(f(x)). �g(x) =. 75 x, f(x) = x – 5 �(g �f )(x) =. 75(x - 5)

Welcome to my domain �The domain of (f �g) is the set of all x in the domain of g in the domain of f. Domain of f Domain of g and domain of f �g

�Example 2. f(x) = x 2 + 2 x and g(x) = 2 x + 1. Find the following. �Find (f �g)(x) Find (g �f )(x)

1. 5 Combinations of Functions Objectives: Students will know how to find arithmetic combinations and compositions of functions.

Arithmetic Combinations of Functions �Let f and g be functions with overlapping domains. Then for all x common to both domains: �(f g)(x) = f(x) g(x) �(fg)(x) = f(x) • g(x) �provided g(x) 0.

Example 1. f(x) = x 2 + 2 x and g(x) = 2 x + 1. Find the following. a) b) c) d)