1. Direct and Inverse Variations
Lesson 9-1
Direct and Inverse Variations 1

2. Direct Variation
y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that
y = kx
The number k is called the constant of variation or the constant of proportionality.

3. Direct Variation
Suppose that y varies directly as x. If y = 5 when x = 30, find the constant of variation and the direct variation equation.
y = kx
5 = k • 30
k =
So the direct variation equation is

4. Example
Suppose that y varies directly as x, and y = 48 when x = 6. Find y when x = 15.
y = kx
48 = k • 6
8 = k
So the equation is y = 8x.
y = 8 ∙ 15
y = 120

5. Inverse Variation
y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that
y = k
The number k is called the constant of variation or the constant of proportionality. x

6. Example
Suppose that y varies inversely as x. If y = 63 when x = 3, find the constant of variation k and the inverse variation equation.
k = 63·3
k = 189
So the inverse variation equation is

7. Powers of x Direct and Inverse Variation as nth Powers of x
y varies directly as a power of x if there is a nonzero constant k and a natural number n such that
y = kxn
y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that

8. Example
At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places.
Translate the problem into an equation.
Substitute the given values for the elevation and distance to the horizon for e and d.
Simplify.
Solve for k, the constant of proportionality.
continued

9. continued So the equation is.
Replace e with 64.
Simplify.
A person 64 feet above the water can see about 9.87 miles.

10. Example
The maximum weight that a circular column can hold is inversely proportional to the square of its height.
If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold.
continued

11. continued So the equation is A 10-foot column can hold 1.28 tons.
Translate the problem into an equation.
Substitute the given values for w and h.
Solve for k, the constant of proportionality.
So the equation is .
Let h = 10.
A 10-foot column can hold 1.28 tons.

12. Direct, Inverse, or Neither

13. Direct or Inverse?
Is the relationship in each table a direct variation, an inverse variation, or neither?
Write functions to model the direct and inverse variations? A. x
0.5 2 6 y
1.5 18

14. Direct or Inverse?
Is the relationship in each table a direct variation, an inverse variation, or neither?
Write functions to model the direct and inverse variations? B. x
0.2
0.6
1.2 y 12 4 2

15. Direct or Inverse?
Is the relationship in each table a direct variation, an inverse variation, or neither?
Write functions to model the direct and inverse variations? C. x 1 2 3 y
0.5

16. Write the Inverse Function
Heart rates and life spans of most mammals are inversely related. Write the inverse function for the table.
Use your function to estimate the average life span of a cat with a heart rate of 126 beats/min.
A squirrel’s heart rate is 190 beats/min. Estimate it’s life span.
An elephant’s life span is about 70 years. Estimate it’s average heart rate.
Mammal
Heart Rate (beats/min)
Life Span (min)
Mouse
634
1,576,300
Rabbit
158
6,307,200
Lion 76
13,140,000
Horse 63
15,768,000

17. Combined Variations
It is possible for three or more variables to be related. When one quantity varies with respect to two or more other quantities, the result is a combined variation.
Combined Variation
Equation Form
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 wy.

18. Writing Combined Variation

19. Finding a Formula

20. Homework: L9-1 (p 499) #8-26e 36-56e