Direct and Inverse
VARIATION
The general equation for DIRECT VARIATION is k is called the constant of variation. We will do an example together.
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.
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.
(b) Find y when x=2 First we find the constant of variation, which was k=8 Now we substitute into y=kx.
Now, YOU try one!! If y varies directly as x and y=6 when x=5, then find y when x=15.
Now lets solve using the equation.
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
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 an equation. Substitute Force = k(length stretched) 30 = k(5)
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 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.
Inverse Variation y varies inversely as x if such that
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
Find y when x=15, if y varies inversely as x and x=10 when y=12 Solve by equation:
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
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 an equation.
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.
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
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