k is called the constant of variation or constant of proportionality.

During a thunderstorm, Peter recorded how long it was between seeing the lightning and hearing the thunder. The distance d in kilometers between Peter and the lightning can be estimated by d=1/3s, where s is the number of seconds between seeing the lightning and hearing the thunder.

Time (sec) Distance(km) The table below shows Peter’s recorded data. Notice how, as time increases, the distance also increases.

If we graph on a coordinate plane we get…

Looking at the graph, what is the slope of the line? Answer: 1/3 Looking at the equation, what is the constant of variation? Answer: 1/3 The constant of variation and the slope are the same!!!!

y varies inversely as x if such that xy=k or For k being the constant of variation or proportionality

Graphically an inverse variation is a curve, not a straight line as in the direct variation.

The graph shows the number of hours, y, it takes x workers to paint a room.

If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation Write the general equation Substitute Solve

(b) Find y when x=2 We found k in part (a): k=8 Now we substitute the value of k into y=kx to get the direct variation formula: y=8x Finally, plug in the given value and solve to find the missing variable:

So lets look at a problem that can by solved by either of these two methods. Another method of solving direct variation problems is to use proportions.

If y varies directly as x and y=6 when x=5, then find y when x=15. Proportion Method: 1.Plug in the given values 2.Solve for the missing variable

Now lets solve using the equation. Either method gives the correct answer, choose the easiest for you.

Now you do one on your own. y varies directly as x, and x=8 when y=9. Find y when x=12. Answer: 13.5

Applications of Direct Variation According to Hook ’ s Law, the force F required to stretch a spring x units beyond its natural length varies directly as x. A force of 30 pounds stretches a certain spring 5 inches. Find how far the spring is stretched by a 50 pound weight.

Set up a proportion Substitute

Now try this problem. Use Hook ’ s Law to find how many pounds of force are needed to stretch a spring 15 inches if it takes 18 pounds to stretch it 13.5 inches. Answer: 20 pounds

If y varies inversely as x, and y=12 when x=10 find: (a) the constant of variation Write the general equation Substitute Solve

(b) Find y when x=15 We found k in part (a): k=120 Now we substitute the value of k into to get the direct variation formula: Finally, plug in the given value and solve to find the missing variable:

Solve by proportion:

Solve this problem using either method. Find x when y=27, if y varies inversely as x and x=9 when y=45. Answer: 15

Applications of Inverse Variation The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle ’ s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

Now try this one on your own. A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle ’ s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

Solving variation word problems. 1. Understand the problem. 2. Write the formula. 3. Identify the known values and substitute in the formula. 4. Solve for the unknown. 5. Make sure to include units of measure

Direct Variation Summary Linear There is no adding or subtracting y is directly proportional to x y varies directly with x Constant of proportionality is k

Direct Variation from a Table To determine direct variation from a table use the form If y divided by x equals the same number throughout the table, that is direct variation and that value is k

Inverse Variation Summary Curve There is no adding or subtracting y is inversely proportional to x y varies inversely with x Constant of proportionality is k

Inverse Variation from a Table To determine direct variation from a table use the form If x times y equals the same number throughout the table, that is inverse variation and that value is k