6.1 - Vertical and Horizontal Shifts

Transcription

1 6.1 - Vertical and Horizontal Shifts Vertical Shifts If y g x is a function and k is a constant, then the graph of y g x k is the graph of y g x y g x k is the graph of y g x Graph f x x, f x x 3, and f x x shifted vertically upward by k units. shifted vertically downward by k units. 3 Horizontal Shifts If y g x is a function and k is a constant, then the graph of y g x k is the graph of y g x y g x k is the graph of y g x Graph f x x, f x x 3, and f x x 3 shifted horizontally to the left by k units. shifted horizontally to the right by k units. A vertical or horizontal shift of the graph of a function is called a translation because it does not change the shape of the graph, but simply translates it to another position in the plane. Shifts or translations are the simplest examples of transformations of a function. Inside and Outside Changes y g x k involves a change to the input value, x, it is called an inside change to g. Similarly, Since since y g x k involves a change to the output value, g x, it is called an outside change. In general, an inside change in a function results in a horizontal change in its graph, whereas an outside change results in a vertical change. For the function y g x, a change inside the function s parentheses can be called an inside change and a change outside the function s parentheses can be called an outside change. Example 7 n f A gives the number of gallons of paint needed to cover a house of area A If meaning of n f A 10 and n f A 10 in the context of painting. ft explain the Example 9 Graph f x x. Define g by shifting the graph of f to the right units and down 1 unit. Find a formula for g in terms of f. 1

2 6.1 Examples: f x 3x x for: 1. Write a formula for the transformation of a. y f x b. y f x r t. Write a formula for the transformation of t a. y r t 1 b. y r t 1 e for: g x and 3. Let f x ln x, g x ln x 5, and hx ln x 5. How do the graphs of compare to the graph of f x? h x 4. The graph of f x contains the point 4, 7 a. f x?. What point must be on the graph of: b. f x? c. f x1 5? g x contains the point 6, 1. Write a formula for a translation of g whose graph contains the point: 6, 3 5. The graph of a. b. 0, 1

3 6. Judging from their graphs, find a formula for gx in terms of f x. 7. Using the graph of gx from #6, sketch a graph of y g x 8. Using the table of f x complete the tables for g and h. x f x x g x 1 g x f x x h x hx f x Review textbook Examples 1,, 3, 4, 5, 6, & 8. 3

4 6. - Reflections and Symmetry In this section we consider the effect of reflecting a function s graph about the x or y-axis. A reflection about the x-axis corresponds to an outside change to the function s formula; a reflection about the y- axis corresponds to an inside change. Reflections For a function f The graph of y f x is a reflection of the graph of y f x The graph of y f x is a reflection of the graph of y f x Example 1 Find a formula in terms of f using the table below for (see text for the graphs): about the x-axis. about the y-axis. a. y g x x f x gx b. y hx c. y k x x f x hx x f x kx Symmetry About the y-axis If f is a function, then f is called an even function if, for all values of x in the domain of f, f x f x. The graph of f is symmetric about the y-axis. Example For the function px x, check algebraically that p p Example 3 for all x. For the function px x, check algebraically that p x px for all x. 4

5 Symmetry About the Origin If f is a function, then f is called an odd function if, for all values of x in the domain of f, f x f x. The graph of f is symmetric about the origin. Example 4 3 For the function qx x, check algebraically that q q for all x. Example 5 3 For the function qx x, check algebraically that q x qx for all x. Example 6 Determine whether the following functions are symmetric about the y-axis, the origin, or neither. 1 3 a. f x x b. g x c. hx x 3x x Example 7 When a yam is taken from a refrigerator at 0 C and put into an oven at 150 C, the yam s temperature rises toward that of the oven. Let Y t be the temperature in C of the yam t minutes after it is put in the oven. Let Dt 150 Y t be the temperature difference between the oven and the yam at time t. See figure 6. in the text which shows a graph of Dt. a. Describe the transformations we apply to the graph of Dt to obtain the graph of Yt. b. Sketch a graph of Yt. c. Explain the significance of the vertical intercept of Y. d. Explain the significance of the horizontal asymptote of Y. 5

6 6.3 - Vertical Stretches and Compressions Formula for Vertical Stretch or Compression If f is a function and k is a constant, then the graph of y k f x is the graph of y f x Vertically stretched by a factor of k, if k 1. Vertically compressed by a factor of k, if 0k 1. Vertically stretched or compressed by a factor k and reflected across the x-axis, if k 0. Example 1 A yam is placed in a 150 C oven. The table below gives values of H r t, the yam s temperature t minutes after being placed in the oven. See figure 6.30 in your text which shows these data points joined by a curve. t, time (min) rt, temperature C a. Describe the function r in words. What do the data tell you about the yam s temperature? b. Make a table of values for qt 1.5r t t, time (min) qt, temperature Graph the function q. C c. How are the functions q and r related? Under what condition might q describe a yam s temperature? 6

7 Stretch Factors and Average Rates of Change Stretching or compressing a function vertically does not change the intervals on which the function increases or decreases. However, the average rate of change of a function, visible in the steepness of the graph, is altered by a vertical stretch or compression. If g x k f x, then on any interval: Average rate of change of g = k (Average rate of change of f ). Example In Example 1, the function function qt 1.5r t H r tgives the temperature C of a yam placed in a 150 C oven. The gives the temperature of the yam placed in a 5 C oven. In both cases, the temperature starts at 0 C. After 10 minutes, r C and q C. a. For each of the two yams, find the rate of change of temperature over ten-minute intervals from t 0 to t 60. Time interval (min) Average rate of change of r Average rate of change of q C min C min b. For each ten-minute interval, what is the relationship between the two values you found in part (a) for the yams? Example Consider the function y f x x, graph the function g x f x 7

8 6.4 - Horizontal Stretches and Compressions Formula for Horizontal Stretch or Compression If f is a function and k a positive constant, then the graph of y f k x Horizontally compressed by a factor of 1 k if k 1. Horizontally stretched by a factor of 1 k if k 1. If 0 k, then the graph of y f k x is the graph of f also involves a horizontal reflection about the y-axis. Example 1 The values of f x are in the table and its graph below. x f x Make a table and a graph of the g x f 1 x. function x gx 8

9 Example Let f x be the function in Example 1. Make a table and a graph for the function hx f x x hx. Review textbook Example 3 Example A carpenter currently builds k chairs per week at a cost of f (k). What do the following expressions represent? a. f k 10 b g b. fbkg 10 c. fbkg d. fbkg 9

10 6.5 Combining Transformations Ordering Horizontal and Vertical Transformations For transformations involving multiple inside and outside changes, it does not matter whether we do the inside changes first, or the outside changes. However, the order of the horizontal changes matters, as does the order of the vertical changes. We can follow the effect of a sequence of transformations on the graph by writing the function in the following form: For nonzero constants A, B, h and k, the graph of the function y A f B x h k is obtained by applying the transformations to the graph of Horizontal stretch/compression by a factor of 1/ B Horizontal shift by h units Vertical stretch/compression by a factor of A Vertical shift by k units f x in the following order: If A < 0, follow the vertical stretch/compression by a reflection about the x-axis. If B < 0, follow the horizontal stretch/compression by a reflection about the y-axis. or y A f B x h k Example 1 The figure shows the graph of a function g. y A f B x h k a. Graph the function that is obtained by first shifting the graph of g horizontally right by 6 units and then compressing horizontally by a factor of 1/3. Give a formula for this function. horizontal shift horizontal stretch / compression g3x 6 Note: 3 1 g x is the same! b. Graph the function that is obtained by first compressing the graph of g horizontally by a factor of 1/3 and then shifting horizontally right by 6 units. Give a formula for this function. horizontal stretch / compression horizontal shift g 3x 6 Note: c. Compare the graphs in your answers to parts (a) and (b). How are they related? 1 g x is the same! 1 10