Direct and Inverse.

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Presentation transcript:

Direct and Inverse

VARIATION

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

(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

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

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

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

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

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

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

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!!!!

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.

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

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.

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

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

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

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.

Step #1: Set up a proportion.

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.

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

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.