Direct Variation

Math IIUnit 7 Functions

Definition: direct variation

A direct variation is a function in the form y = kx where k does not equal 0.

When we talk about a direct variation, we are talking about a relationship where as x increases, y increases or as

x decreases, y decreaeses at a CONSTANT RATE.

Direct Variation Model

The two variables x and y are said to vary directly if their relationship is:

y = kx

k is the same as m (slope) k is called the constant of variation

Is an Equation a Direct Variation?If it is, find the constant of variation.

5x + 2y = 0 Solve for y.1. Subtract 5x.2. Divide by 2.

Yes, it’s a direct variation.Constant of variable, k, is

y 5

2x

5

2.

Is an Equation a Direct Variation?If it is, find the constant of variation.

5x + 2y = 9Solve for y.

1. Subtract 5x.

2. Divide by 2.

No, it’s not a direct variation.It’s not in the form y = kx.

y 9 5x2

y 4.5 5

2x

The price of hot dogs varies directly with the number of hotdogs you buy

You buy hotdogs. x represents the number of hotdogs you buy.y represents the price you pay.

y = kx Let’s figure out k, the price per hotdog. Suppose that when you buy 7 hotdogs, it costs $21.

Plug that information into the model to solve for k.

y = kx21 = k(7) Now divide both sides by 7 to solve for k. 7 7

k = 3 The price per hotdog is $3.

y = 3x You could use this model to find the price (y) for any number of hotdogs (x) you buy.

x

y

The graph of y = 3xgoes through the origin.

All direct variation graphs go through the origin, because when x = 0, y= 0 also.

x (number of hotdogs)

y (price)

y = 3x

(1,3) When you buy 1 hotdog, you pay $3..

(2,6) When you buy 2 hotdogs, you pay $6

. (3,9) When you buy 3 hotdogs, you pay $9

.(0,0) When you buy 0 hotdogs, you pay $0

Finding the Constant of Variation (k)

STEPS

1. Plug in the known values for x and y into the model: y = kx

2. Solve for k3. Now write the model y = kx and

replace k with the number4. Use the model to find y for other

values of x if needed

ExampleThe variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

Example

84 (24)

y kx

k

The variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

1.

Example

84 (24)

y kx

k

84 (24)

24 247

2

k

k

The variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

1.

Example

84 (24)

y kx

k

84 (24)

24 247

2

k

k

The variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

1.

7

2y x

Example

84 (24)

y kx

k

84 (24)

24 247

2

k

k

7

2y x

The variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

1. 2.

7

2y x

Example

84 (24)

y kx

k

84 (24)

24 247

2

k

k

7

2y x

The variables x and y vary directly. When x = 24, y = 84.

1. Write the direct variation model that relates x and y.

2. Find y when x is 10.

710

235

y

y

1. 2.

7

2y x When x = 10, y = 35

ExampleThe variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

Example

118

2

y kx

k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

Example

118

2

y kx

k

1(2)18 (2)

2k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

Example

118

2

y kx

k

1(2)18 (2)

2k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

36 k

Example

118

2

y kx

k

1(2)18 (2)

2k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

36 k36y x

Example

118

2

y kx

k

1(2)18 (2)

2k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

36 k

2.

36y x

36y x

Example

118

2

y kx

k

1(2)18 (2)

2k

The variables x and y vary directly. When x = ½, y = 18.

1. Write the direct variation model that relates x and y.

2. Find y when x is 5.

1.

36 k

2.

36y x

36y x

36(5)

180

y

y

When x = 5, y = 180

Real-World Example

Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see

the lightning, you are about 2 miles from the lightning. Write an equation for the

relationship between time and distance.

Real-World Example Continued….

Relate: The distance varies directly with the time. When x = 10, y = 2.

Define: Let x = number of seconds between seeing lightning and hearing thunder.Let y = distance in miles from lightning.

Real-World Example Continued…

y = kx Use general form of direct variation.2 = k(10) Substitute 2 for y and 10 for x.

Solve for k.

Write an equation using the value for k.

k 1

5

y 1

5x

ExampleA recipe for a dozen corn muffins calls for 1 cup of flower. The number of muffins varies directly with the amount of flour you use. Write a direct

variation for the relationship between the number of cups of flour and the number of

muffins.

Example

The force you must apply to lift an object varies directly with the object’s weight. You would need to apply 0.625 lb of force to a windlass to lift a 28-lb weight. How much

force would you need to lift 100 lb?

Example:

A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

Example:

Julio wages vary directly as the number of hours that he works. If his wages for 5 hours are $29.75, how much will they be for 30 hours?

Inverse Variation

Math IIUnit 7 Functions

Inverse Variation

Inverse is very similar to direct, but in an inverse relationship as

one value goes up, the other goes down. There is not necessarily a constant rate.

Definition: inverse variation

An inverse variation is a function in the form y = k

x where k does not equal 0

where k is still the constant of variation.

Graphing Inverse Variations

An inverse variation can also be identified by its graph. Some inverse variation graphs are shown. Notice that each graph has two parts that are not connected.

Also notice that none of the graphs contain (0, 0). In other words, (0, 0) can never be a solution of an inverse variation equation.

ExampleThe variables x and y vary inversely. When x = -12, y = 0.5.

1. Write the inverse variation model that relates x and y.

2. Find y when x is 3.

ExampleIf y varies inversely as x and x = 18 when y = 6, find y when x = 8.

Example

If y varies inversely with x and y = 12 when x = 2, find y when x = 8.

Example

Y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

Example

The cost per person to rent a mountain cabin is inversely proporitonal (varies inversely) to the number of people who share the rent. If the cost is $36 per person when 5 people share, what is the cost per person when 8 people share?

Example

The time it takes to fly from Los Angeles to New York varies inversely as the speed of the place. If the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h?

Example

The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm3 under pressure of 32 kg/cm2. What will be its volume under pressure of 40 kg/cm2?

Joint VariationMath II

Unit 7 Functions

Inverse Variation

There are situations when more than one variable is involved in a direct variation problem. In these cases the problem is called joint variation. The equation remains the same except that additional variables are included in the product. For example if z varies jointly with the values of x and y the equation will be

z = kxy .

Example:

Variable I varies jointly as the values of P and T.

If I = 1200 when P = 5000 and T = 3 , find I when P = 7500 and T = 4.

Example:

The cost c of materials for a deck varies jointly with the width w and the length l.

If c = $470.40 w hen w = 12 and l = 16 , find the cost when w = 10 and l = 25.

Example:

The value of real estate V varies jointly with the neighborhood index N and the square footage of the house S. If V = $376, 320 when N = 96 and S = 1600 , find the value of a property with N = 83 and S = 2150.

Example:

The number of gallons g in a circular swimming pool varies jointly with the square of the radius r2 and the depth d. If g = 754 when r = 4 and d = 2 , find the number of gallons in the pool when r = 3 and d = 1.5.

EXAMPLE*Joint – Direct AND Inverse!

To build a sound wall along the highway, the amount of time t needed varies directly with the number of cement blocks c needed and inversely with the number of workers w.

A sound wall made of 2400 blocks, using six workers takes 18 hours to complete. How long would it take to build a wall of 4500 blocks with 10 workers?

EXAMPLE*Joint – Direct AND Inverse!

The number of hours needed to assemble computers varies directly as the number of computers and inversely as the number of workers. If 4 workers can assemble 12 computers in 9 hours, how many workers are needed to assemble 48 computers in 8 hours?

EXAMPLE*

An egg is dropped from the roof of a building. The distance it falls varies directly with the square of the time it falls. If it takes 4 seconds for the egg to fall 15 feet, how long will it take the egg to fall 500 feet?