THE RECIPROCAL FUNCTION FAMILY AND RATIONAL FUNCTIONS AND THEIR GRAPHS L E S S O N 9-2 A N D L E S S O N 9-3

Transcription

1 THE RECIPROCAL FUNCTION FAMILY AND RATIONAL FUNCTIONS AND THEIR GRAPHS L E S S O N 9-2 A N D L E S S O N 9-3

2 ASSIGNMENT 2/12/15 Section 9-2 (p506) 2, 6, 16, 22, 24, 28, 30, 32 section 9-3 (p513) 1 18

3 Functions that model inverse variations belong to a family whose parent is the reciprocal function f x = 1 x, where x 0. Transformations f x f x = a x = a x + k moves up k moves down f x = a moves right x h f x = a moves left x+h a is the stretch (if a > 1) or shrink if 0 < a < 1 a < 0 is a reflection in the x-axis

4 GRAPHING AN INVERSE VARIATION Sketch a graph of y = 6, x 0 x What are the asymptotes? Each part of the graph is called a branch.

5 GRAPHING AN INVERSE VARIATION Sketch a graph of y = 3, x 0 x What are the asymptotes? Each part of the graph is called a branch.

6 GRAPHING RECIPROCAL FUNCTIONS Draw the graph of y = 4 Describe the transformations x

7 GRAPHING RECIPROCAL FUNCTIONS Draw the graph of y = 2 Describe the transformations x

8 A musical pitch is determined by the frequency of vibration of the sound waves reaching the ear. The greater the frequency, the higher is the pitch. Frequency is measured in vibrations per second, or hertz (Hz). The pitch (y) produced by a panpipe varies inversely with the length (x) of the pipe. 564 Write the function: y x Find the length of the pipe that produces a pitch of 277 Hz. Pitches of 247 Hz and 370 Hz. Find the length of pipes that will produce each pitch. The asymptotes of this equation are y=0 and x=0. Explain why this makes sense in terms of the panpipe. Desmos Graphing Calculator

9 GRAPHING TRANSLATIONS OF RECIPROCAL FUNCTIONS Graph on desmos y = 1 x, y = 1 x 1, y = 1 x+2 What are the vertical and horizontal asymptotes for each graph? How do the vertical asymptotes relate to the denominators equaling zero? Now graph y = 1 x, y = 1 x + 1, and y = 1 x 2 What are the vertical and horizontal asymptotes for each graph?

10 GRAPHING A TRANSLATION Sketch the graph of y = 1 x 2 3

11 GRAPHING A TRANSLATION Sketch the graph of y = 1 x+7 3

12 GRAPHING A TRANSLATION Sketch the graph of y = 7 x 8 4

13 WRITING THE EQUATION OF A TRANSFORMATION Write an equation for the translation of y = 5 x has asymptotes at x = 2 and y = 3. that

14 WRITING THE EQUATION OF A TRANSFORMATION Write an equation for the translation of y = 1 x that has asymptotes at x = 4 and y = 5.

15 WRITING THE EQUATION OF A TRANSFORMATION Write an equation for the translation of y = 13 x has asymptotes at x = 5 and y = 8. that

16 SECTION 9-3 Objective: Students will identify properties of rational functions

17

18 The graph is continuous because it has no jumps, breaks, or holes in it. It can be drawn with a pencil that never leaves the paper. There is no real value of x that makes the denominator 0

19 The graph is not continuous since there is a break X can not be 2 and -2

20 The graph is not continuous. X can not be -1

21 A point of discontinuity is either a hole or a vertical asymptote.

22 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y 3( x 4) 2 x x 12

23 For each rational function, find any points of discontinuity. a. y = 3 x 2 x 12 The function is undefined at values of x for which x 2 x 12 = 0. x 2 x 12 = 0 Set the denominator equal to zero. (x 4)(x + 3) = 0 Solve by factoring or using the Quadratic Formula. x 4 = 0 or x + 3 = 0 Zero-Product Property x = 4 or x = 3 Solve for x. There are points of discontinuity at x = 4 and x = 3.

24 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y x x 1

25 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y x x 2 1 1

26 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y x x

27 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y ( x x 2 4) 16

28 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y 2x( x 5) x 2 25

29 FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY. y ( x 1)( x 2 x 2x 4) 8

30 Describe the any points of discontinuity ( vertical asymptotes and holes for the graph of each rational function). a. y = x 7 (x + 1)(x + 5) Since 1 and 5 are the zeros of the denominator and neither is a zero of the numerator, x = 1 and x = 5 are vertical asymptotes.

31 b. y = (x + 3)x x is a zero of both the numerator and the denominator. The graph of this function is the same as the graph y = x, except it has a hole at x = 3.

32 c. y = (x 6)(x + 9) (x + 9)(x + 9)(x 6) 6 is a zero of both the numerator and the denominator. The graph of the function is the same as the graph y = 1 (x + 9) which has a vertical asymptote at x = 9, except it has a hole at x = 6.,

33 DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL FUNCTION). y ( x x 1 1)( x 2)

34 DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL FUNCTION). y ( x 2)( x 1) ( x 2)

35 DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL FUNCTION). y ( x x 2 1)( x 2)