Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 LINES AND FUNCTIONS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.4 TRIGONOMETRIC FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.1 Suppose that f and g are functions with domains D1 and D2, respectively. The functions f + g, f − g and f · g are defined by © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.1 Combinations of Functions © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.1 Combinations of Functions The domain of f is the entire real line and the domain of g is the set of all x ≥ 1. The domain of both ( f + g) and (3 f − g) is © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.1 Combinations of Functions The domain of f is the entire real line and the domain of g is the set of all x ≥ 1. The domain is . © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.2 Finding the Composition of Two Functions © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.2 Finding the Composition of Two Functions Note that for x to be in the domain of g, we must have x ≥ 2. The domain of f is the whole real line, so this places no further restrictions on the domain of f ◦ g. The domain of ( f ◦ g) is © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.2 Finding the Composition of Two Functions The resulting square root requires x2 − 1 ≥ 0 or |x| ≥ 1. Since the “inside” function f is defined for all x, the domain of g ◦ f is © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.4 Vertical Translation of a Graph © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.4 Vertical Translation of a Graph © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS Vertical Translations In general, the graph of y = f (x) + c is the same as the graph of y = f (x) shifted up (if c > 0) or down (if c < 0) by |c| units. We usually refer to f (x) + c as a vertical translation (up or down, by |c| units). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 13

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.5 A Horizontal Translation Compare and contrast the graphs of y = x2 and y = (x − 1)2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 14

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.5 A Horizontal Translation © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 15

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.5 A Horizontal Translation To avoid confusion on which way to translate the graph of y = f (x), focus on what makes the argument (the quantity inside the parentheses) zero. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS Horizontal Translations In general, for c > 0, the graph of y = f (x − c) is the same as the graph of y = f (x) shifted c units to the right. Likewise (again, for c > 0), you get the graph of y = f (x + c) by moving the graph of y = f (x) to the left c units. We usually refer to f (x − c) and f (x + c) as horizontal translations (to the right and left, respectively, by c units). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 17

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.7 Comparing Some Related Graphs Compare and contrast the graphs of y = x2 − 1, y = 4(x2 − 1) and y = (4x)2 − 1. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 18

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.7 Comparing Some Related Graphs © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 19

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS 5.7 Comparing Some Related Graphs © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 20

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS Scaling the x- and y-axes Based on Example 5.7, notice that to obtain a graph of y = cf (x) for some constant c > 0, you can take the graph of y = f (x) and multiply the scale on the y-axis by c. To obtain a graph of y = f (cx), for some constant c > 0, you can take the graph of y = f (x) and multiply the scale on the x-axis by 1/c. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 21

TRANSFORMATIONS OF FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS Summary © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 22