9 1 Inverse Variation VARIATION Direct Direct Variation

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: DAYS X=1 SNOW LEVEL(Y) 3 X=2 X=3 6 9 • This is a direct variation because the model can be represented by y = 3 x

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: DAYS X=1 STORE PROFIT(Y) 15 X=2 X=3 7. 5 5 • This is an inverse variation because the model can be represented by y = 15/x

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 X y 0. 5 1. 5 2 6 6 18 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 = 3 x Ask: What is going on? X y 2 4 6 y increases as x decreases. However, when you multiply x by y you get different values: 3. 2 1. 6 1. 1 Since there is no clear relationship, we say that there is no variation!

X y 0. 8 0. 6 0. 4 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 Direct Variation Part This is the Inverse 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 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. • That means that hs=1, 000, 000 • Let’s find out your heart rate

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

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

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