9.1 – Inverse Variation

VARIATION – Direct Direct Variation is… A relation or function that can be represented by y =kx where k is a constant. For example: This is a direct variation because the model can be represented by y = 3x DAYS X = 1 X = 2 X = 3 SNOW LEVEL(Y) 3 6 9

VARIATION – Direct EX Y varies directly with X. Y = 100 when x =5. Find k. Then find x when y = 150

VARIATION – Direct EX – Y varies directly with X. Y = 100 when x =5. Find k. Then find x when y = 150

INVERSE VARIATION Inverse Variation is… A relation or function that can be represented by xy = k where k is a constant; or y = k/x For example: This is an inverse variation because the model can be represented by y = 15/x DAYS X = 1 X = 2 X = 3 STORE PROFIT(Y) 15 7.5 5

INVERSE VARIATION EX Y varies inversely with X. Y = 30 when x =4. Find k. Then find x when y = 150

INVERSE VARIATION EX – Y varies inversely with X. Y = 30 when x =4. Find k. Then find x when y = 150

Identifying Direct and Inverse Variation Ask: What is going on? y increases as x increases. Since y is three times as big as x each time, this is direct variation; y = 3x X 0.5 2 6 y 1.5 18 Ask: What is going on? y increases as x decreases. However, when you multiply x by y you get different values: X 2 4 6 y 3.2 1.6 1.1 Since there is no clear relationship, we say that there is no variation!

We have inverse variation. Therefore, the equation is y = 0.72 x 0.8 0.6 0.4 y 0.9 1.2 1.8 Ask: What is going on? y increases as x decreases. Test: is it an inverse variation? What is k? We have inverse variation. Therefore, the equation is y = 0.72 x

Joint variation Joint variation is variation with more than 2 variables (more than x and y) EX y varies directly with x and inversely with z. This would be  y= kx z This is the Inverse Variation Part This is the Direct Variation Part

Joint variation EX – Y varies directly with x and inversely with z. Y = 100 when x = 5 and z = 4. Find k. Then find x when y = 200 and z = 10

Joint variation EX – Y varies directly with x and inversely with z. Y = 100 when x = 5 and z = 4. Find k. Then find x when y = 200 and z = 10

More examples: translate the following: Y varies directly with the square of x Y varies inversely with the cube of x Z varies jointly with x and y Z varies jointly with x and y and inversely with w Z varies directly with x and inversely with the product of w and y

Application Application – heart rates and life spans of mammals are inversely related. Let h = heart rate (bpm) and s = life span (min). The constant, k, is 1,000,000,000. That means that hs=1,000,000,000 Let’s find out your heart rate

Application: Working in groups, find the approximate lifespan of each mammal HEART RATE Bpm LIFE SPAN In minutes Mouse 634 1,576,800 = 3 years Rabbit 158 6,307,200 = 12 years Lion 76 13, 140,000 = 25 years Horse 63 15,768,000 = 30 years Reminder: heart rates and life spans of mammals are inversely related and k = 1,000,000,000

Graphing Inverse Variation y = k x Graphs of inverse functions will look something like the cross between a linear graph and a parabolic curve. In this case, we are just looking at the graph in the first quadrant. When we look at a true inverse variation function, there will always be two graphs to the functions, diagonal from each other. We will look more at these graphs later on in this chapter.

Graphing Inverse Functions with a Graphing Calculator 1. Press MODE. Scroll down and highlight the word DOT.

Graphing Inverse Functions with a Graphing Calculator Let’s try 2. Press Y= and enter the function 12/x.

Graphing Inverse Functions with a Graphing Calculator 3. Graph the function. Practice: Graph