1. NOTES 1-1C & D: PROPERTIES DIRECT & INVERSE (INDIRECT) VARIATION
OBJECTIVE: Student should be able to…
Identify and describe a variable and expression
Evaluate an expression
Identify properties and evaluate to a given property
TAKS OBJECTIVE 1: The student will describe functional relationships in a variety of ways.

2. NOTES 1-1C & D: PROPERTIES DIRECT & INVERSE (INDIRECT) VARIATION
Commutative Property:
the “order” changes
3 + 7 = 7 + 3
4 x 5 = 5 x 4
Associative Property:
the “groups” change
(2 + 4) + 6 = 2 + (4 + 6)
5 = 5
(3 x 5) x 7 = 3 x (5 x 7)
the numbers in a sum or difference are BOTH multiplied by the same value.
Distributive Property:
4( 3 + 2) =
4 x x 2
3( 5 – 2) =
3 x 5 – 3 x 2

3. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
There are two types of variation: _________________
and ________________.
Direct variation means that as x _____________, y
_____________, or as x __________________, y
_________________. (The more hours you work, the more money you earn.)
Inverse variation means that as x _________________, y
___________________. (The more miles you drive, the less gas in your tank.)
DIRECT
INVERSE
increases
increases
decreases
decreases
increases
decreases
or vise versa

4. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
Both direct and inverse variation have constants of variation (k).
EXAMPLE 1 Study the table below. Determine the type of
variation, and tell how you would find the constant
of variation.
Direct Variation
Hours Worked 5 11 14 17
Total Pay
30.00
66.00
84.00
102.00 x y
Constant = 6

5. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
General Equation for direct variation:
EXAMPLES: Find the value indicated.
2. If y varies directly as x, and y = 6 when x = 8, find y when
x = 12.
3. The force required to stretch a spring, F, varies directly with the amount the spring is stretched, s. Ten pounds is needed to stretch a spring 8 inches. How many pounds would be needed to stretch the spring 32 inches?
6(12) = 8(y)
72 = 8y
9 = y
8(x) = 10(32)
Direct Variation
8x = 320
x = 40

6. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
4. The distance driven, d, varies directly with the time traveled, t. If you have driven 175 miles for 5 hours, how long would you drive for 210 miles?
Direct Variation
175(t) = 5(210)
175t = 1050
t = 6

7. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
EXAMPLE 5 Study the table below. Determine the type of
variation, and tell how you would find the constant
of variation.
Indirect Variation
Miles 18 20 30 60
Gallons of Gas 10 9 6 3 x y
18(10)
20(9)
30(6)
60(3)
180
180
180
180
Constant = 180

8. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
General Equation for inverse variation:
k = xy
EXAMPLES: Find the value indicated.
6. If y varies inversely as x and y = 6 when x = 2, find x when
y = 4
(2)(6) = 4x
12 = 4x
3 = x

9. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
EXAMPLES: Find the value indicated.
7. The volume, V, of a gas at constant temperature varies inversely with the pressure, P. When the volume is 100 cubic inches, the pressure is 25 pounds. Find the volume when the pressure is 50 pounds.
Inverse (Indirect) Variation
k = VP
100(25) = V(50)
2500 = 50V
50 = V

10. NOTES 1-1D: DIRECT & INVERSE (INDIRECT) VARIATION
8. The number of slices, n, cut from a bread loaf of constant length varies inversely as the uniform thickness, t, of each slice. When there are 16 slices, each slice is 15 mm thick. Find the number of slices when the thickness is 12 mm.
Inverse (Indirect) Variation
k = nt
16(15) = n(12)
240 = 12n
20 = n