5.1 ­ Direct Variation Jan 25.notebook

When two variables are related, there are many possible factors that affect the relationship. There are two types of relationships that we are going to examine in this unit. The first is a DIRECT VARIATION. In a direct variation, one variable equals a fixed multiple of the other. This means that there is only one factor that affects the value of the other variable.

Example 1: Trevor earns $8 for every hour in his part time job. a) Complete the chart and graph the results. y 40

Amount of Time(hr) Earnings ($)

0 1 2

32

Earnings ($)

Direct Variation

January 25, 2017

24

16

8

3 4

x 0

1

2

3 Time (hr)

4

5

6

Constant of Variation (m)

In a direct variation, it is the ratio (fraction) of the corresponding values of the variables. For example, if y varies directly as x, then the constant of variation is given by

b) What is the constant of variation for Trevor since his amount of earnings varies directly as time.

The graph can be used as a model to make predictions about the relationship between the variables. Interpolation is using a graph to estimate values INSIDE, or between graphed data. Extrapolation is using a graph to estimate values OUTSIDE or beyond graphed data. c) Using your graph, estimate how long Trevor must work to make $36.

d. What are the two variables in this example? Equations can be created to generate a general statement about how the two variables are related. For direct variations, a general equation can be created using y = mx, where y varies directly as x. y is always the dependent variable and x is always the independent variable.

e. State the equation for the amount that he earns.

f) Use your equation to find out how long must he work to earn $220?

5.1 ­ Direct Variation Jan 25.notebook

January 25, 2017

Example 2: Given that y varies directly as x, find the constant of variation. a. y = 45, x = 9 b. x = -2, y = 8

Example 3: Susan can jog at a steady pace of 150 m per minute. a. Identify the independent and dependent variable.

b. Complete a table of values for the first 5 minutes that Susan is running. Time (t) Distance(d)

c. y = 3, x = 4

d. Find the constant of variation.

c. Graph the relationship. y 800 700 600 500

e. Write the equation describing the relationship between the two variables.

400 300 200 100 x 0

1

2

3

4

5

6

7

8

In general for direct variation:

• y varies directly with x • •

y = mx

• The graph is a straight line that passes through the origin.

f. What does this graph and Example 1's graph have in common?