Direct, Inverse and Joint Variation

Direct, Inverse and Joint Variation

What is 'Variation'?

• Variation is a relationship between 2 or more situations (variables) that have an unchanged constant of variation (k)

• Examples:

• The way grades are related to the hours of study

• The way temperature is related to attendance at a baseball game

• The way hours worked is related to the amount of a paycheck

• There are 3 main types: Direct, Inverse and Joint

3 Types of Variation models:

Direct:   $y=kx$  

Inverse:   $y=\frac{k}{x}$  

Joint:   $y=kxz$  (like Direct but has another variable)

Direct Variation:

• The graph is a line that passes through (0, 0)

• k is the constant of proportionality and the slope of the line

• As one variable increases or decreases the other does the same

Examples of Direct Variation

As one thing increases or decreases, the other does the same thing!

Inverse Variation

Graphs look like a rational function.

As one variable increases or decreases, the other does the opposite!

Inverse Variation Examples

Notice as one thing increases or decreases , the other does the opposite!

Joint Variation

Same as direct but includes the product of another variable.

Ex. The number of bacteria in mayonnaise depends on temperature in kitchen AND time left out of fridge

Direct Variation

As one variable increases (x), the other also increases (y) and vice versa.
Use model:  $y=kx$  

Let's work through a problem:

If y varies directly as x and y=6 when x=11, find y when x =3.

Basically, we will use the direct model twice---once to find k, and another to find the solution.

First, write the equation model:

• If y varies directly as x and y=6 when x=11, find y when x =3.

• The direct model here is: y = kx

Substitute and solve for k:

• If y varies directly as x and y=6 when x=11, find y when x =3.

• Substitute 6 and 11 into the model (y=kx) and solve for k:

• 6 = k (11)

• k = 6/11

Rewrite the model substituting k and the other number:

• If y varies directly as x and y=6 when x=11, find y when x =3.

• Model: y = kx

• To find y , substitute k = 6/11 and x = 3

• Plug these into the model to find y:

• y = (6/11)(3) = 18/11

Let's try another example

Multiple Choice

If m varies directly as p, and m = 35 when p = 5, find m when p is 6.

Which shows the correct model ?

• m = kp

• p = km

• m = k/p

Multiple Choice

If m varies directly as p, and m = 35 when p = 5, find m when p is 6. The model is m = kp.

Now substitute and solve for 'k'.

• k =5

• k = 1/7

• k = 7

• k = 175

Multiple Choice

If m varies directly as p, and m = 35 when p = 5, find m when p is 6.

Find m:

Substitute the value of k you found (k=7) and p =6 into the model and solve for m:

• m = 42

• m = 36

• m = 25

• m = 49

Okay, let's do these same steps with a word problem...

You don't have to use x and y...you can use other variables if it makes more sense to you.

Break up the problem into 3 parts: write model, substitute to find k, then substitute again and solve.

Multiple Choice

Trina's paycheck earnings varies directly as the number of hours she works. If she works 19 hours and earns $187.15, what should she earn if she worked 40 hours?

Which is the correct model if 'p' is for paycheck earnings and 'h' is for hours worked ?

• p = k/h

• h = kp

• p = kh

Multiple Choice

Trina's paycheck earnings varies directly as the number of hours she works. If she works 19 hours and earns $187.15, what should she earn if she worked 40 hours?

Find k by substituting:

Use the model: p = kh (h = 19 and p = 187.15)

• k = 1/9

• k = 8.72

• k = 19

• k = 9.85

Multiple Choice

Trina's paycheck earnings varies directly as the number of hours she works. If she works 19 hours and earns $187.15, what should she earn if she worked 40 hours?

Find the paycheck earnings (p) by substituting:

k = 9.85 and h = 40 into the model: p =kh

• $394

• $443.25

• $472.80

• $512.20

Multiple Choice

Try this one all the way through...

The gas pressure in a chamber varies directly with the temperature in the chamber. If the pressure in the chamber is 150 atmospheres (atm) when the chamber is at 50 degrees, what is the pressure in the chamber when the temperature of the chamber is 75 degrees?

Use the 3 steps: write model, substitute to find k, then substitute again and solve.

• 175 atm

• 200 atm

• 225 atm

• 275 atm

The other variations work the same way...

• The problem will usually tell you which variation to use

• Write out the variation model equation based on the information given before and after the word 'direct', 'inverse' or 'joint'.

• Substitute the first set of values given to find k

• Plug in k and the other values given to find the final answer

Inverse Variation:

• Use the model:    $y=\frac{k}{x}$  

• Follow the same steps as we did with Direct:

• Write the model---use different letters if you wish

• Find k---use the 1st 2 values they give

• Solve the rest--use 'k' and the last value given

Multiple Choice

The width of a rectangle varies inversely with its length. If the width is 6 (y) when the length is 24 (x), find the length when the width of the rectangle is 4.

What is the model to be used?

• y = kx

• y = k/x

• x = ky

Multiple Choice

The width of a rectangle varies inversely with its length. If the width is 6 (y) when the length is 24 (x), find the length when the width of the rectangle is 4.

Now find 'k' and write the new model.

• y = 4x

• y = 144x

• y = 4/x

• y = 144/x

Multiple Choice

The width of a rectangle varies inversely with its length. If the width is 6 (x) when the length is 24 (y), find the length when the width of the rectangle is 4.

Now finish solving. (Hint: recall you found 'k' to be 144)

• length = 4

• length = 6

• length = 36

• length = 42

Multiple Choice

Try this all the way through.....

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

• 5.33 hours

• 6.75 hours

• 12 hours

• 10 hours

Joint variation problem:

• Just like direct but includes an extra variable

• model is  $y=kxz$  

Multiple Choice

Suppose y varies jointly as x & z. If y = -180 when z = 15 and x = -3,

then find y when x = 7 and z = -5.

Solve like you've done previously but use this model: y=kxz

• y = -140

• y = 140

• y = -4

• y = 4

Combined Variation

• Sometimes problems will combine direct and inverse together

• Create a model knowing that 'direct' means multiply and 'inverse' means divide

• Ex. If y varies directly with x and inversely with z......    $y=\frac{kx}{z}$ 

Multiple Choice

Suppose f varies directly as g, and f varies inversely as h. If f = -12 when h = 4 and g = -3, find g when f = 28 and h = 8.

Which model is the correct set up?

• f = kgh

• f = kg/h

• f = k/h

• f = kg

Multiple Choice

Use the model you found to solve:

Suppose f varies directly as g, and f varies inversely as h. If f = -12 when h = 4 and g = -3, find g when f = 28 and h = 8.

Use the 3 steps: write model, substitute to find k, then substitute again and solve.

• g = 14

• g = -14

• g = 16

• g = -16