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. Direct Variation
When we talk about a direct variation, we are talking about a relationship where as x increases, y increases or decreases at a CONSTANT RATE.

5. If y varies directly as x, and y=24 and x=3 find y when x=2.
(a) First, you will need to find the constant of variation (k).
Write the general equation
Substitute-you know y=24 when x=3.

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

7. Now, YOU try one!!
If y varies directly as x and y=6 when x=5, then find y when x=15.

8. Now lets solve using the equation.

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

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

11. 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!!!!

12. 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.

13. Set up an equation. Substitute Force = k(length stretched) 30 = k(5)

14. 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

15. Inverse Variation
Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down. There is not necessarily a constant rate.

16. Inverse Variation
y varies inversely as x if
such that

17. Inverse Variation
With Direct variation we Multiply our k’s and x’s. y=kx
In Inverse variation we will Divide them. y=k/x

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

19. Solve this problem on your own!
Find x when y=27, if y varies inversely as x and x=9 when y=45.
Answer: 15

20. 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.

21. Step #1: Set up an equation.

22. 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.

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

24. 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.

25. Joint Variation is when one quantity varies directly as the product of two or more quantities. So, just like direct, but with more than one variable.
y=kxz

26. Combined is when you use more
than one variation in one problem.
For example if y varies directly
with x and inversely with z, it would
be set up as y=(kx)/z