1. Direct
and
Inverse

2. VARIATION

3. The general equation for DIRECT VARIATION is
k is called the constant of variation.
We will do an example together.

4. (a) Find the constant of variation
If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation
(b) Find y when x=2
(a) Find the constant of variation
Write the general equation
Substitute

5. (b) Find y when x=2
First we find the constant of variation, which was k=8
Now we substitute into y=kx.

6. Another method of solving direct variation problems is to use proportions.
Therefore...

7. So lets look at a problem that can by solved by either of these two methods.

8. If y varies directly as x and y=6 when x=5, then find y when x=15.
Proportion Method:

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

10. 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

11. What does the graph y=kx look like?
A straight line with a y-intercept of 0.

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

13. We will apply what we know and try this problem.
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.

14. Set up a proportion
Substitute

15. 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

16. Inverse Variation
y varies inversely as x if
such that xy=k or
Just as with direct variation, a proportion can be set up solve problems of indirect variation.

17. A general form of the proportion
Lets do an example that can be solved by using the equation and the proportion.

18. Find y when x=15, if y varies inversely as x and x=10 when y=12
Solve by equation:

19. Solve by proportion:

20. 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

21. Lets apply what we have learned.
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.

22. Step #1: Set up a proportion.

23. 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.

24. What does the graph of xy=k look like? Let k=5 and graph.

25. This is a graph of a hyperbola.
Notice: That in the graph, as the x values increase the y values decrease.
also
As the x values decrease the y values increase.