1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Transformations of Functions SECTION Learn the meaning of transformations. Use vertical or horizontal shifts to graph functions. Use reflections to graph functions. Use stretching or compressing to graph functions.

3 © 2010 Pearson Education, Inc. All rights reserved TRANSFORMATIONS If a new function is formed by performing certain operations on a given function f, then the graph of the new function is called a transformation of the graph of f.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing Vertical Shifts Let Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing Vertical Shifts Solution Make a table of values.

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Graphing Vertical Shifts Solution continued Graph the equations. The graph of y = |x| + 2 is the graph of y = |x| shifted two units up. The graph of y = |x| – 3 is the graph of y = |x| shifted three units down.

7 © 2010 Pearson Education, Inc. All rights reserved VERTICAL SHIFT Let d > 0. The graph of y = f (x) + d is the graph of y = f (x) shifted d units up, and the graph of y = f (x) – d is the graph of y = f (x) shifted d units down.

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Functions for Horizontal Shifts Let f (x) = x 2, g(x) = (x – 2) 2, and h(x) = (x + 3) 2. A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide. Describe how the graphs of g and h relate to the graph of f.

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Functions for Horizontal Shifts

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Functions for Horizontal Shifts

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Functions for Horizontal Shifts All three functions are squaring functions. Solution The x-intercept of f is 0. The x-intercept of g is 2. a.g is obtained by replacing x with x – 2 in f. For each point (x, y) on the graph of f, there will be a corresponding point (x + 2, y) on the graph of g. The graph of g is the graph of f shifted 2 units to the right.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Writing Functions for Horizontal Shifts Solution continued The x-intercept of f is 0. The x-intercept of h is –3. b.h is obtained by replacing x with x + 3 in f. For each point (x, y) on the graph of f, there will be a corresponding point (x – 3, y) on the graph of h. The graph of h is the graph of f shifted 3 units to the left. The tables confirm both these considerations.

13 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL SHIFT The graph of y = f (x – c) is the graph of y = f (x) shifted |c| units to the right, if c > 0, to the left if c < 0.

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Sketch the graph of the function Solution Step 1 Identify and graph the known function Choose Graphing Combined Vertical and Horizontal Shifts

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Step 2 Identify the constants d and c in the transformation g (x) = f (x – c ) + d. Step 3 Since c = –2 < 0, the graph of is the graph of f shifted horizontally two units to the left. Graphing Combined Vertical and Horizontal Shifts Solution continued d

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solution continued Graphing Combined Vertical and Horizontal Shifts Step 4 Because d = –3 < 0, graph by shifting the graph of three units down.

17 © 2010 Pearson Education, Inc. All rights reserved REFLECTION IN THE x -AXIS The graph of y = – f (x) is a reflection of the graph of y = f (x) in the x-axis. If a point (x, y) is on the graph of y = f (x), then the point (x, –y) is on the graph of y = – f (x).

18 © 2010 Pearson Education, Inc. All rights reserved REFLECTION IN THE x -AXIS

19 © 2010 Pearson Education, Inc. All rights reserved REFLECTION IN THE y -AXIS The graph of y = f (–x) is a reflection of the graph of y = f (x) in the y-axis. If a point (x, y) is on the graph of y = f (x), then the point (–x, y) is on the graph of y = f (–x).

20 © 2010 Pearson Education, Inc. All rights reserved REFLECTION IN THE y -AXIS

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Combining Transformations Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|. Solution Step 1Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|.

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Combining Transformations Solution continued Step 2Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Combining Transformations Solution continued Step 3Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Stretching or Compressing a Function Vertically Solution Sketch the graphs of f, g, and h on the same coordinate plane, and describe how the graphs of g and h are related to the graph of f. Let x–2–1012 f(x)f(x)21012 g(x)g(x)42024 h(x)h(x)11/20 1

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Stretching or Compressing a Function Vertically Solution continued

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Stretching or Compressing a Function Vertically Solution continued The graph of y = 2|x| is the graph of y = |x| vertically stretched (expanded) by multiplying each of its y–coordinates by 2. The graph of |x| is the graph of y = |x| vertically compressed (shrunk) by multiplying each of its y–coordinates by.

27 © 2010 Pearson Education, Inc. All rights reserved VERTICAL STRETCHING OR COMPRESSING The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is 1.A vertical stretch away from the x-axis if a > 1; 2. A vertical compression toward the x-axis if 0 < a < 1. If a < 0, the graph of f is first reflected in the x-axis, then vertically stretched or compressed.

28 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL STRETCHING OR COMPRESSING The graph of y = f (bx) is obtained from the graph of y = f (x) by multiplying the x-coordinate of each point on the the graph of y = f (x) by and leaving the y-coordinate unchanged. The result is 1.A horizontal stretch away from the y-axis if 0 < b < 1; 2. A horizontal compression toward the y-axis if b > 1.

29 © 2010 Pearson Education, Inc. All rights reserved HORIZONTAL STRETCHING OR COMPRESSING If b < 0, first obtain the graph of f (|b|x) by stretching or compressing the graph of y = f (x) horizontally. Then reflect the graph of y = f (|b|x) in the y-axis.

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Stretching or Compressing a Function Horizontally The graph of a function y = f (x) is given. No formula for f is given. Sketch the graph of each of the related functions.

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Stretching or Compressing a Function Horizontally Solution Stretch the graph of y = f (x) horizontally by a factor of 2. Each point (x, y) transforms to (2x, y).

32 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Stretching or Compressing a Function Horizontally Solution continued Compress the graph of y = f (x) horizontally by a factor of. Each point (x, y) transforms to

33 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Stretching or Compressing a Function Horizontally Solution continued Reflect the graph of y = f (2x) in the y-axis. Each point (x, y) transforms to (–x, y).

34 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Combining Transformations Sketch the graph of the function f (x) = 3 – 2(x – 1) 2. Solution Step 1y = x 2 Identify a related function. Step 2y = (x – 1) 2 Shift right 1.

35 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Combining Transformations Solution continued Step 3y = 2(x – 1) 2 Stretch vertically by a factor of 2. Step 4y = –2(x – 1) 2 Reflect in x-axis.

36 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Combining Transformations Solution continued Step 5y = 3 – 2(x – 1) 2 Shift three units up.

37 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Poiseuille’s Law for Arterial Blood Flow For an artery with radius R, the velocity v of the blood flow at a distance r from the center of the artery is given by where c is a constant that is determined for a particular artery.

38 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Poiseuille’s Law for Arterial Blood Flow We write v = v(r), or v(r) = c(R 2 – r 2 ). Starting with the graph of y = r 2, sketch the graph of y = v(r), given that the artery has radius R = 3 and c =10 4. Solution

39 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Poiseuille’s Law for Arterial Blood Flow Solution continued Transform y = v(r) = r 2 y =10 4 r 2 Stretch vertically by 10 4 y = –10 4 r 2 Reflect in x-axis y = 9 ∙10 4 – 10 4 r 2 Shift 9 ∙10 4 units up

40 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Poiseuille’s Law for Arterial Blood Flow The velocity of the blood decreases from the center of the artery (r = 0) to the artery wall (r = 3), where it ceases to flow. The portions of the graphs corresponding to negative values of r do not have any physical significance, but it is somewhat easier to work with the familiar graph of y = r 2. Solution continued