5-1 Rate of Change and Slope

Quick Review

Rate of change shows the relationship between two changing quantities. The slope of a line is the ratio of the vertical change (the rise) to the horizontal change (the run).

slope = rise run = y 2 − y 1 x 2 − x 1 slope , equals . fraction rise , over run end fraction . equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction

The slope of a horizontal line is 0, and the slope of a vertical line is undefined.

Example

What is the slope of the line that passes through the points (1, 12) and (6, 22)?

slope = y 2 − y 1 x 2 − x 1 = 22 − 12 6 − 1 = 10 5 = 2 slope , equals . fraction y sub 2 , minus , y sub 1 , over x sub 2 , minus , x sub 1 end fraction . equals . fraction 22 minus 12 , over 6 minus 1 end fraction . equals , 10 over 5 , equals 2

Exercises

Find the slope of the line that passes through each pair of points.

6. (2, 2), (3, 1)
7. (4, 2), (0, 2)
8. ( − 1 , 2 ) , ( 0 , 5 ) open negative 1 comma 2 close comma open 0 comma 5 close
9. ( − 3 , − 2 ) , ( − 3 , 2 ) open negative 3 comma negative 2 close comma open negative 3 comma 2 close

Find the slope of each line.

5-2 Direct Variation

Quick Review

A function represents a direct variation if it has the form y = k x , y equals k x comma  where k ≠ 0 . k not equal to 0 .  The coefficient k is the constant of variation.

Example

Suppose y varies directly with x, and y = 15 bold italic y equals 15  when x = 5 . bold italic x equals 5 .  Write a direct variation equation that relates x and y. What is the value of y when x = 9 ? bold italic x equals 9 question mark

y = k x Start with the general form of a direct variation . 15 = k ( 5 ) Substitute  5  for  x  and  15  for  y . 3 = k Divide each side by  5  to solve for  k . y = 3 x Write an equation . Substitute  3  for  k  in  y = k x . table with 4 rows and 3 columns , row1 column 1 , y , column 2 equals k x , column 3 cap startwiththegeneralformofadirectvariation . . , row2 column 1 , 15 , column 2 equals k , open 5 close , column 3 cap substitute . 5 , for , x , and , 15 , for , y . , row3 column 1 , 3 , column 2 equals k , column 3 cap divideeachsideby . 5 . tosolvefor . k . , row4 column 1 , y , column 2 equals 3 x , column 3 cap writeanequation . .cap substitute . 3 , for , k , in , y equals k x . , end table

The equation y = 3 x y equals 3 x  relates x and y. When x = 9 , y = 3 ( 9 ) , x equals 9 comma y equals 3 open 9 close comma  or 27.

Exercises

Suppose y varies directly with x. Write a direct variation equation that relates x and y. Then find the value of y when x = 7 . bold italic x equals 7 .

12. y = 8  when  x = − 4 . y equals 8 , when , x equals negative 4 .
13. y = 15  when  x = 6 . y equals 15 , when , x equals 6 .
14. y = 3  when  x = 9 . y equals 3 , when , x equals 9 .
15. y = − 4  when  x = 4 . y equals negative 4 , when , x equals 4 .

For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation.

16. x	y
    − 1 negative 1	− 6 negative 6
    2	3
    5	12
    9	24

17. x	y
    − 3 negative 3	− 7.5 negative 7.5
    − 1 negative 1	2.5
    2	− 5 negative 5
    5	− 12.5 negative , 12.5

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