1. Direct
and
Inverse

2. VARIATION

3. Do Now
A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

4. Using Direct Variation Method to solve this word problem:
Step 1: Find points in table
Word Problem:
A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?
Step Three: Use the equation
to find the unknown.
400 =36.25x
or x = 11.03
Step Two: Find the constant variation and equation:
k = y/x or k = 290/8 or 36.25
y = x

5. Recall,
The general equation for DIRECT VARIATION is
k is called the constant of variation.

6. Is this a direct variation
Is this a direct variation? If yes, give the constant of variation (k) and the equation.
Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x

7. The k values are different!
Is this a direct variation? If yes, give the constant of variation (k) and the equation.
No!
The k values are different!

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

9. Direct Variation Method:

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

11. If y varies directly as x, and y=24 and x=3 find: Find y when x=2
First find the constant of variation
Write the general equation
Substitute

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

13. You can also solve it using the Proportion Method.

14. How does the graph y=kx look like?
What is the constant of variation?

15. Tell if the following graph is a Direct Variation or not.
Yes No No
Yes

16. Word problem 2.
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.

17. Set up a proportion
Substitute

18. 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

19. Practice
Worksheet: 1-10

20. Exit Question 1. Which of the following is a direct variation?
2. Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52.

21. Do Now
Given that y varies directly with x, and y = 6 when x=-5,
Find y when x = -8.

22. y varies directly as x, and x=8 when y=9. Find y when x=12.
Answer: 13.5

23. Tell if the following graph is a Direct Variation or not.
Yes No No No

24. 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.

25. Tell whether each relationship is an inverse variation. Explain.1 2

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

27. 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

28. 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.

29. The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law
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.

30. What does the graph of xy=k look like?

31. 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.

32. Practice
Textbook p / 1, 2, 5, 6
Hw: p. 128/ 7, 9