7.4 RECIPROCAL FUNCTIONS
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1 7.4 RECIPROCAL FUNCTIONS x
2 VOCABULARY Word Know It Well Have Heard It or Seen It No Clue RECIPROCAL FUNCTION ASYMPTOTE VERTICAL ASYMPTOTE HORIZONTAL ASYMPTOTE
3 RECIPROCAL a mathematical expression or function so related to another that their product is one; the quantity obtained by dividing the number one by a given quantity. Reciprocal of 2 is = 2 Reciprocal of 3 4 is = 4 3 Reciprocal of x is = x Reciprocal of f(x) is = f(x)
4 Compare the Graphs of a Function and Its Reciprocal Investigate (make a table of value and sketch the graphs) the graphs of y= f(x) and its reciprocal function y=, where f(x) = x. Examine f(x) how the functions are related.
5
6 y = x x = 0 is a nonpermissible value Value 0 is NOT is the Range of the funcion (y nac NEVER be 0)
7 Compare the Graphs of a Function and Its Reciprocal Investigate (make a table of value and sketch the graphs) the graphs of y= f(x) and its reciprocal function y=, where f(x) = x. Examine f(x) how the functions are related.
8
9 ASYMPTOTE a line whose distance from a given curve approaches zero
10 ASYMPTOTE a line whose distance from a given curve approaches zero
11 ASYMPTOTE a line whose distance from a given curve approaches zero
12 VERTICAL ASYMPTOTE for reciprocal functions, occur at the nonpermissible values of the function
13 VERTICAL ASYMPTOTE the line x = a is a vertical asymptote if the curve approaches the line more and more closely as x approaches a, and the values of the function increase or decrease without bound as x approaches a
14 HORIZONTAL ASYMPTOTE describes the behaviour of a graph when x is very large
15 HORIZONTAL ASYMPTOTE the line y = b is a horizontal asymptote if the values of the function approach b when x is very large
16 y = x x = 0 is a VERTICAL ASSYMPTOTE y = 0 is a HORIZONTAL ASSYMPTOTE
17 What are the Equations VERTICAL and/or HORIZONTAL ASYMPTOTES?
18 a) The reciprocal function is y = 2x + 5 b) The vertical asymptote occurs at any non permissible value of x: 2x + 5 = 0 2x = -5 x = -5/2
19 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) y = 2x + 5
20 For INVARIANT POINTS y = 2x + 5 y = 2x + 5 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5)
21 For INVARIANT POINTS = 2x = 2x + 5 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5)
22 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) -2 INVARIANT POINT y = 2x + 5
23 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) INVARIANT POINT INVARIANT POINT y = 2x + 5
24 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) INVARIANT POINT INVARIANT POINT y = 2x + 5
25 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) /5 = -0.2 INVARIANT POINT INVARIANT POINT y = 2x + 5
26 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) /5 = INVARIANT POINT INVARIANT POINT y = 2x + 5
27 x = -5/2 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(2x +5) /5 = /5 = 0.2 INVARIANT POINT INVARIANT POINT y = 2x + 5
28
29
30 a) The reciprocal function is y = 3x 9 b) The vertical asymptote occurs at any non permissible value of x: 3x - 9 = 0 3x = 9 x = 3
31 x = 3 y = 3x 9 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(3x - 9)
32 INVARIANT POINTS - = 3x 9 x = 3 = 3x 9 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(3x - 9) 8/ INVARIANT POINT
33 INVARIANT POINTS - = 3x 9 x = 3 = 3x 9 c) Sub in several more values for x (make sure there are some x values from either side of the vertical asymptote + INVARIANT POINTS): x y = /(3x - 9) 8/ / INVARIANT POINT INVARIANT POINT
34 x = 3 y = 3x 9 x y = /(3x - 9) 8/ / INVARIANT POINT INVARIANT POINT
35 x = 3 y = 3x 9 x y = /(3x - 9) 8/ / INVARIANT POINT INVARIANT POINT
36 x = 3 y = 3x 9 x y = /(3x - 9) 8/ / INVARIANT POINT INVARIANT POINT
37 x = 3 y = 3x 9 x y = /(3x - 9) 8/ / INVARIANT POINT INVARIANT POINT
38
39
40 a) The reciprocal function is b) Non-permissible values of x occur when the denominator of the corresponding rational expression is equal to 0. The reciprocal function is undefined at these values, so its graph has vertical asymptotes with equations x = 2 and x = -2.
41 c) To find the x-intercepts, let y = 0. There is no value of x that makes this equation true. Therefore, there are no x-intercepts.
42 c) To find the y-intercept, let x = 0.
43 d) Without DESMOS The coordinates of the vertex are: The x-intercepts occur at: Draw the asymptotes: Plot the invariant points:
44 d) Without DESMOS Plot the invariant points:
45 d) Without DESMOS The coordinates of the vertex are: The x-intercepts occur at: Draw the asymptotes: Plot the invariant points:
46 The coordinates of the vertex are: The x-intercepts occur at: Draw the asymptotes: Plot the invariant points:
47 The coordinates of the vertex are: The x-intercepts occur at: Draw the asymptotes: Plot the invariant points:
48
49 PAGE: PROBLEMS #:,3, 6, 7, 9
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