LESSON 2-2 NOTES:
DIRECT VARIATION
EXAMPLE: Joe makes $9 an hour working at a local coffee shop. Complete the chart below.
Hours
Worked
Amount
Earned
1
2
3
4
5
6
7
Write the data from the chart above as ordered pairs (hours worked, amount earned):
{(
,
), (
,
), (
,
), (
,
), (
,
), (
,
), (
,
)}
Is the above relation a function?
What happens to the amount earned as the number of hours worked increases?
Plot the ordered pairs on the
coordinate plane.
Some quantities, such as hours worked and amount earned in the above example, are in a
relationship where the ratio of corresponding values is constant. In the example above, the amount
earned increases directly as the number of hours worked increases. This type of relationship is
called direct variation. Direct variation functions can be written as y = kx, or k = (k 0), where
k is the constant of variation.
In the above example, the direct variation function is: y = 9x, where x = hours worked and
y = amount earned. The constant of variation, k, is k = = 9.
EXAMPLES/PRACTICE: IDENTIFY DIRECT VARIATION FROM TABLES.
For each function, determine whether y varies directly with x.
If so, what is the constant of variation k?
1)
2)
3)
x
y
x
y
x
y
-6
-2
2
8
1
5
9
3
3
12
2
10
21
7
5
20
3
14
EXAMPLES/PRACTICE: IDENTIFYING DIRECT VARIATION FROM EQUATIONS
For each function, determine whether y varies directly with x.
If so, what is the constant of variation k?
4) 5y = 15x
5) 4y = 12x + 8
In direct variation, the ratio
So
=
6) 3x + 2y = 0
is the same for all pairs of data where x ≠ 0.
is true for the ordered pairs (x1, y1) and (x2, y2), where neither x1 nor x2 is 0.
EXAMPLES/PRACTICE: USING A PROPORTION TO SOLVE A DIRECT VARIATION
1) Suppose y varies directly with x, and y = 12 when x = -4. What is y when x = 24?
2) Suppose y varies directly with x, and y = 12 when x = 36. What is x when y = 7?
3) Suppose y varies directly with x, and y = 15 when x = 3. What is y when x = 12?
EXAMPLES/PRACTICE:
USING DIRECT VARIATION TO SOLVE A PROBLEM
1) A salesperson's commission varies directly with sales. In other words, a person's
commission increases as the amount of sales increases. For $1000 in sales, the
commission is $75. What is the commission for $3500?
a) Method 1 - Using a direct variation function
b) Method 2 - Using a proportion
Step 1: Find the value of k
Step 2: Write the direct variation function
2) A new car has a 15-gallon gas tank. On one tank of gas, the car can travel 360 miles.
The number of miles traveled varies directly with the number of gallons of gas the
car uses.
a) Write an equation (direct variation function) that relates the number of miles
traveled with the number of gallons of gas used. (Hint: Find k.)
b) Use your equation from part (a) to determine the number of miles the car can
travel on 7 gallons of gas?