11-6 Inverse Variation

Quick Review

When the product of two variables is constant, the variables form an inverse variation. You can write an inverse variation in the form xy = k or y = k x , y equals , k over x . comma  where k is the constant of variation.

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Example

Suppose y varies inversely with x, and y = 8 when x = 6. What is an equation for the inverse variation?

x y = k General form of an inverse  variation 6 ( 8 ) = k Substitute  6  for  x  and  8  for  y . 48 = k Simplify . x y = 48 Write an equation . table with 4 rows and 3 columns , row1 column 1 , x y , column 2 equals k , column 3 cap generalformofaninverse . variation , row2 column 1 , 6 open 8 close , column 2 equals k , column 3 cap substitute . 6 , for , x , and , 8 , for , y . , row3 column 1 , 48 , column 2 equals k , column 3 cap simplify , . , row4 column 1 , x y , column 2 equals 48 , column 3 cap writeanequation . . , end table

Exercises

Suppose y varies inversely with x. Write an equation for the inverse variation.

27. y = 7 when x = 3
28. y = 4 when x = 2.5
29. y = − 9 y equals negative 9  when x = 2
30. y = 5 when x = − 5 x equals negative 5

Graph each inverse variation.

31. xy = 15
32. y = − 18 x y equals , negative 18 over x
33. Running Suppose a runner takes 45 min to run a route at 8 mi/h at the beginning of training season. By the end of training season, she can run the same route in 38 min. What is her speed at the end of training season?

11-7 Graphing Rational Functions

Quick Review

A rational function can be written in the form f ( x ) = polynomial polynomial . f , open x close , equals . fraction polynomial , over polynomial end fraction . .  The graph of a rational function in the form y = a x − b + c y equals . fraction eh , over x minus b end fraction . plus c  has a vertical asymptote at x = b and a horizontal asymptote at y = c. A line is an asymptote of a graph if the graph gets closer to the line as x or y gets larger in absolute value.

Example

What is the graph of f ( x ) = 1 x − 1 + 2 ? f , open x close , equals . fraction 1 , over x minus 1 end fraction . plus 2 . question mark

From the form of the function, you can see that there is a vertical asymptote at x = 1 and a horizontal asymptote at y = 2. Sketch the asymptotes.

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Make a table of values. Then graph the function.

Exercises

Identify the excluded value for each function.

34. f ( x ) = 5 x f , open x close , equals , 5 over x
35. y = 3 x + 4 y equals . fraction 3 , over x plus 4 end fraction

Identify the asymptotes of the graph of each function. Then graph the function.

36. y = 1 x + 2 y equals . fraction 1 , over x plus 2 end fraction
37. f ( x ) = − 2 x + 3 f , open x close , equals . fraction negative 2 , over x plus 3 end fraction
38. y = 5 x − 4 + 1 y equals . fraction 5 , over x minus 4 end fraction . plus 1
39. f ( x ) = 3 x − 5 − 1 f , open x close , equals . fraction 3 , over x minus 5 end fraction . minus 1
40. Physics For a 225-watt bulb, the intensity I of light in lumens at a distance of x feet is I = 225 x 2 . i equals . fraction 225 , over x squared end fraction . .
    1. What is the intensity of light 5 ft from the bulb?
    2. Suppose your distance from the bulb doubles. How does the intensity of the light change? Explain.

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