Direct and Inverse Variation

Lesson on Direct and Inverse Variation

Please make sure to write notes as you are reviewing each slide.

Some text here about the topic of discussion

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

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

​Ok, let's try a few questions

Multiple Choice

Is the given table a direct variation? If so, what is the constant?

• Yes; 1/4
• Yes; 4
• Yes; 2
• No

Multiple Choice

Is the given table a direct variation? If so, what is the constant of variation?

• Yes; 1/3
• Yes; 3
• Yes; 5
• No

Multiple Choice

Is the given table a direct or inverse variation? What is the constant of variation?

• Direct, 12
• Inverse, 12
• Direct, 1/12
• Inverse, 1/12

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

​Ok, let's try a few questions

Multiple Choice

Y varies directly to x. If y = 16 when x = 4, which equation represents this situation?

• $$y=\frac{1}{4}x$$

• $$y=12x$$

• $$y=4x$$

• $$y=64x$$

Multiple Choice

 Y varies directly with x, and y is 84 when x is 16. Which equation represents this situation?

• y=1344x
• y=100x
• y=5.25x
• y=4/21x

​Example 2:

Y varies directly with x, and y = 24 when x = 2. Find y when x = 8

1. First write the formula: "y varies directly" → y = kx

2. Substitute the values given → 24 = k(2)

3. Solve for "k" AKA "Constant of Variation" → 24/2 = k → 12 = k

4. ​Substitute our "new" k value → y = 12(x)

5. Now solve for y when x = 8 → y = 12(8) → y = 96

6. Done!​

Let's try two more

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

​Now we move onto the applications of Direct variation in "real world" situations

Examples of Direct Variation

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

​Karen earns $28.50 for working 6 hours. If the amount m she earns varies directly with h the number of hours she works, how much will she earn for

working 10 hours?

​

• ​Important parts to notice:

  1. "$28.50 for working 6hrs"

  2. "amount m she earns varies directly with h the number of hours"

• Equations to use: y = k(x)

• The set up: 28.50 = K(6)

• Solve for k: 28.50/6 = k → ​4.75

• Substitute: y = 4.75x

• Solve for 10 hours: y = 4.75(10) → y = $47.50

• Done!​

​Example 2 :

The number of gallons g of fuel used on a trip varies directly with the number of miles m traveled. If a trip of 270 miles required 12 gallons of fuel, how many gallons are required for a trip of 400 miles?​

​​Important parts to notice:

1. "270 miles required 12 gallons of fuel"

2. "amount g (gallons) fuel used varies directly with the number of miles m traveled"

• Equations to use: y = k(x)

• The set up: 12 = K(270)

• Solve for k: 12/270 = k → ​0.04444444

• Substitute: y = 0.0444444x

• Solve for gallons: y = 0.044444(400) → y = 17.78 gallons

• Done!​

​Ok, let's try a few questions

Multiple Choice

The amount of money spent at the gas station varies directly with the number of gallons purchased. When 11.5 gallons of gas were purchased the cost was $37.72. Write the equation?

• $$37.72=k⋅11.5$$

• $$37.72=11.5\cdot k​$$

• $$m=3.28⋅g$$

• $$m=\frac{433.78}{g}$$

Multiple Choice

Ounces of medication vary directly with the weight of the patient. If a 120-pound adult requires 3 ounces of a medication, then how many ounces would be needed for a 200-pound adult?

• 2

• 3

• 5

• 9

Multiple Choice

The Height of a wave in California varies directly with the seconds that pass by.  At 4 seconds, the wave is 6 feet high.  How many seconds will give you a wave that is 10 feet high? 

• y = 2/3x
• 6
• 6.6
• 7

​Ok

Now we move onto Inverse Variation​

Inverse Variation

Graphs look like a rational function.

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

​Ok, let's try a few questions

Multiple Choice

If "y varies inversely as x," then what is the variation equation of the given table of values?

• $$x=45y$$  

• $$y=45x$$  

• $$y=\frac{45}{x}$$  

• $$y=\frac{0.2}{x}$$  

Multiple Choice

Is the given table a direct or inverse variation? What is the constant of variation?

• Direct. k = 12

• Inverse. k = 12

• Direct. k = 1/12

• Inverse. k = 1/12

Multiple Choice

What relationship does the table represent?

• Direct

• Inverse

• Neither

Let's work through an example:

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

1. First write the formula: "y varies inversely" → y = k/x

2. Substitute the values given → 40 = k/(16)

3. Solve for "k" AKA "Constant of Variation" → 40*16 = k → 640 = k

4. ​Substitute our "new" k value → y = 640/(x)

5. Now solve for x when y = -5 → -5 = 640/x → x = 640/-5 → -128

6. Done!​

​

​Let's try one more

​y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

1. First write the formula: "y varies inversely" → y = k/x

2. Substitute the values given → 7 = k/(-4)

3. Solve for "k" AKA "Constant of Variation" → 7*-4 = k → -28 = k

4. ​Substitute our "new" k value → y = -28/(x)

5. Now solve for y when x = 5 → y = -28/5 → -5.6

6. Done!​

​

​

​Ok, let's try a few questions

Multiple Choice

Given that y varies inversely with x, if x =7 and y = 4, what is y when x = 2?

• 14
• 10
• 2
• -14

Multiple Choice

If Y varies inversely with X, and Y = 60 when X = 3, find Y when X = 18.

• 180

• 20

• 57

• 10

​Now onto the applications!

☻​

Inverse Variation Examples

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

​Example 1

The time it takes to travel a fixed distance varies inversely with the speed traveled. If it takes Pam 40 minutes to bike to her fishing spot at 9 miles

per hour, how long will it take her if she rides at 12 miles per hour?​

​​Important parts to notice:

1. "40 minutes at 9 miles per hour"

2. "time it takes to travel a fix distance varies inversely with speed"

• Equations to use: y = k/(x)

• The set up: 40 = K/(9)

• Solve for k: 40*9 = k → ​360

• Substitute: y = 360/x

• Solve for time when at 12mph: y = 360/(12) → y = 30 minutes

• Done!​

​Example 2

The time to prepare a field for planting is inversely proportional to the number of people who are working. A large field can be prepared by 5

workers in 24 days. In order to finish the field sooner, the farmer plans to hire additional workers. How many workers are needed to finish the field

in 15 days?​

1. "5 workers in 24 days"

2. "time it takes to prepare a field is inversely proportional to the number of people"

3. Equations to use: y = k/(x)

4. The set up: 24 = K/(5)

5. Solve for k: 24*5 = k → ​120

6. Substitute: y = 120/x

7. Solve for workers (x) when days are 15: 15 = 120/(x) → x = 120/15 → 8

8. Done!​

​

Open Ended

The force, F newtons, exerted by a magnet on a metal object is inversely proportional to the square of the distance d cm.

When d = 2 cm, F = 50 N.

a) Express F in terms of d.

b) Find the force when the distance between the magnet and metal object is 10cm.

c) Find the distance between the magnet and metal object when the force is 8N.

Multiple Choice

Mrs. Ralls will pay $5000 to move to her new house. The amount each mover earns, A, varies inversely with the amount of movers, m, that will move her into her new place.

Which equation best represents this scenario?

• A = 5000 + m

• 5000 = A + m

• 5000 = Am

• A = 5000m

Open Ended

The time taken, t seconds, that it takes a water heater to boil water is inversely proportional to the power, p watts, of the water heater. When P = 2000W, T = 252 seconds.

Find the time it takes to boil water when P = 800W

You probably noticed, but just in case

• Direct: An increase (decrease) in one quantity produces an increase (decrease) in the other

• Inverse: An increase (decrease) in one quantity produces a decrease (increase) in the other.

• k is named ​"Constant of Variation" or "Constant of Proportionality"

• When it comes to points if ​you have: (6,1) the inverse is (1,6)

• Direct variation graph always crosses the (0,0) AKA "origin point"​

​That is it for notes: make sure to complete your assignment on the next step!