Section 2-1Direct Variation

r Varies Directly as c:

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation:

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

Direct Variation Function:

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0

Can also be known as “directly proportional”

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0

Can also be known as “directly proportional”***The cost of gas varies directly as the amount of

gas purchased

r Varies Directly as c: When r gets larger, so does c; When r gets smaller, so does c

Constant of Variation: k is a nonzero constant in y = kxn, and n is a positive integer

Direct Variation Function: A function of the form y = kxn with k ≠ 0 and n > 0

Can also be known as “directly proportional”***The cost of gas varies directly as the amount of

gas purchasedThe more you get, the more it costs

Example 1

Rewrite the statement, “The cost of gas varies directly as the amount of gas purchased.”

Example 1

Rewrite the statement, “The cost of gas varies directly as the amount of gas purchased.”

“The cost of gas is directly proportional to the amount of gas purchased.”

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.P = kE

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.P = kE

b. Identify the independent and dependent variables.

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.P = kE

b. Identify the independent and dependent variables.

Independent: E

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.P = kE

b. Identify the independent and dependent variables.

Independent: E Dependent = P

Example 2

The weight of an object on planet P varies directly with its weight on Earth E.

a. Write an equation relating P and E.P = kE

b. Identify the independent and dependent variables.

Independent: E Dependent = Pk is just a constant

Example 3

The ingredients for a pizza and the price are proportional to its area. This means the quantity of

ingredients is proportional to the square of its radius. Suppose a pizza 12 in. in diameter costs $7.00. If the price varies directly as the square of its radius, what would a pizza 16 in. in diameter cost? What

about an 18 in. pizza?

Example 3

Example 3

c = cost

Example 3

c = cost r = radius

Example 3

c = cost r = radiusc = kr2

Example 3

c = cost r = radiusc = kr2

7 = k(6)2

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36)

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64)

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64) ≈ $12.44

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64) ≈ $12.44

c = 736 (9)

2

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64) ≈ $12.44

c = 736 (9)

2 = 736 (81)

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64) ≈ $12.44

c = 736 (9)

2 = 736 (81) = $15.75

Example 3

c = cost r = radiusc = kr2

7 = k(6)2 7 = k(36) k = 736 c = 7

36 r2

c = 736 (8)

2 = 736 (64) ≈ $12.44

c = 736 (9)

2 = 736 (81) = $15.75

A 16 in. diameter pizza would cost $12.44 and an 18 in. diameter pizza would cost $15.75.

Steps to solving a direct variation problem

Steps to solving a direct variation problem

1.Write an equation to describe the variation

Steps to solving a direct variation problem

1.Write an equation to describe the variation

2.Find k

Steps to solving a direct variation problem

1.Write an equation to describe the variation

2.Find k3.Rewrite the function using k

Steps to solving a direct variation problem

1.Write an equation to describe the variation

2.Find k3.Rewrite the function using k4.Evaluate

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

32 = k(2)

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

32 = k(2)

k = 16

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

32 = k(2)

k = 16

y = 16x

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

32 = k(2)

k = 16

y = 16x

y = 16(5)

Example 4

Find k if y varies directly as x where y = 32 when x = 2. Then find y when x = 5.

y = kx

32 = k(2)

k = 16

y = 16x

y = 16(5) = 80

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn48 = k(12)

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn48 = k(12)

k = 4

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn48 = k(12)

k = 4m = 4n

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn48 = k(12)

k = 4m = 4n

m = 4(3)

Example 5

m is directly proportional to n. If m = 48 when n = 12, find m when n = 3.

m = kn48 = k(12)

k = 4m = 4n

m = 4(3) = 12

Homework

Homework

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