Direct and Inverse VariationSOL A8by Robert Lotze, Moody Middle School

Direct Variation The longer you shower, the more water

you use.

You can describe thisRelationship using

Direct Variation

The Direct Variation Formula Y=kx (where k ≠ zero)

Does this look familiar??????

The national average for time in a shower is 12.2 minutes. The average shower head uses 6 gallons of water per minute. This means the average shower uses 73.2 gallons of water. How much water is this in a year for you? (how often do you shower?)

Here is a table to help you.X (minutes

3 6 9 12

Y(gallons 18 36 72 90A relationship exists between the number of minutes In ashower, and the gallons of water used.

The equations for this table is: y = 6x

This is called a direct variation. Thismeans that as x increases, y increases,Or, as x decreases, y decreases

Another exampleEmma’s wages vary directly as the number of hours she works. If her wages for 5 hours are $29.75, how much will she be paid for 30 hours?

Frist, find Emma’s hourly pay. Let x = number of hours worked, and let y = Emma’s pay. The value of k is the amount of money Emma is paid per hour. This is called the constant of the direct variation.

divide both sides by

Emma is paid $5.95 per hour. Thus the constant of this direct variation “k” is $5.95. Now plug this into the equation .

You can also use proportions to solve problems of direct variation.

For the last problem:

Your turn1. Steve drove 330 miles using 15 gallons of fuel. At the

same rate, how many gallons of fuel will Steve need to plan on using if he has to drives from Richmond to Salt Lake City, about 1650 miles?

So: thus

Steve will need plan on using 75 gallons for his trip.

Your turn2. If varies directly as , and when , find when

Answer:

Inverse VariationThe length of a violin stringvaries inversely as the frequencyof its vibration. In other words, theshorter the string the higher the pitch. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8 inch string.

You can describe this relationship using Inverse Variation

Inverse VariationAn inverse variation is describedby an equation of the form:

So with the violin string:

The constant of the inverse variation is 5120

So, since we can find the frequency of any length of string.

Shortening the string to 8 inches we get:

So, the frequency of an 8 inch string is 640 cycles per second.

Your turnIf you have ever seen or been on a seesaw, you will know that the heavier person has to sit closer to the fulcrum (pivot point) of the see saw to balance. This is a type of lever. This is also an inverse variation.

The fulcrum is placed in the middle of a 20-foot seesaw. Chloe, who weighs 120 lbs., is seated 9 feet from the fulcrum. How far from the fulcrum should Anthony sit if he weighs 135 lbs.?

Remember for inverse variation.So: or

Anthony needs to sit 8 feet from the fulcrum.

Your TurnAssume that varies inversely as .

If the value of is 90 when the value of is 6, find the value of when the value of is 45.

So: or

In Summary Direct Variation: All equations of lines of Direct Variation

pass through the origin. Both variables Increase or Decrease

Inverse Variation: When one variable increases, the other

decreases.