Direct and Inverse

VARIATION

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

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 36.25 36.25 or x = 11.03 Step Two: Find the constant variation and equation: k = y/x or k = 290/8 or 36.25 y = 36.25 x

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

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

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!

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

Direct Variation Method:

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

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

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

You can also solve it using the Proportion Method.

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

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

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.

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

Practice Worksheet: 1-10

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.

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

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

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

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.

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

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

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

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.

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.

What does the graph of xy=k look like?

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.

Practice Textbook p. 127- 128/ 1, 2, 5, 6 Hw: p. 128/ 7, 9