1. 9.1 – Inverse Variation

2. 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

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

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

5. 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

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

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

8. 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!

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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.

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

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

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