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

Assignment p. 168 # 1, 4, 6

Objectives: Students will know how to identify and graph shifts, reflections, and nonrigid transformations of functions. Extra Practice Extra Examples standards-test-algebra-2

The Original Six Constant Function f(x) = c Identity Function f(x) = x XY XY 00 11

Absolute Value Function f(x) = |x| Square Root Function XY XY 00 42

Quadratic Function f(x) = x 2 Cubic Function f(x) = x 3 XY XY

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 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)Horizontal shift c units to the right Y 2 = (x – 4) 2 h(x) =f(x + c)Horizontal shift c units to the left Y 3 = (x + 4) 2 h(x) =f(x) - cVertical shift c units downward Y 4 = x 2 – 4 h(x) =f(x) + cVertical shift c units upward Y 5 = x 2 + 4

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 eflecting_functions eflecting_functions Register me as your coach and do 10 problems

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 Reflected in the ???-axis. b) h(x) = -(x 3 + 3) = -x Reflected in the ???-axis.

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

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 – 3x 2. Because 3 – 3x 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. Xf(x)=1- x 2 g(x) = 3 – 3x

XY -2(8/3) (2/3) (8/3) XY XY XY

Please describe the following function g(x) = -2f(x) Reflection? Wider or Narrower? 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 $ 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) =.75x (f  g)(x) = f(g(x)) = [.75x] - 5 The composition of the function g with the function f is (g  f )(x) = g(f(x)). g(x) =.75x, 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 + 2x and g(x) = 2x + 1. Find the following. Find (f  g)(x) Find (g  f )(x)

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