9.5 = variation functions. direct variation direct variation – y varies directly as x

9.5 = Variation Functions

Direct Variation

Direct Variation – y varies directly as x

Direct Variation – y varies directly as x

y = kx

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Inverse Variation

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Inverse Variation – y varies inversely as x

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Inverse Variation – y varies inversely as x

y = k

x

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Inverse Variation – y varies inversely as x

y = k

x

Joint Variation

Direct Variation – y varies directly as x

y = kx

*Note: k = constant of variation (a #)

Inverse Variation – y varies inversely as x

y = k

x

Joint Variation – y varies jointly as x and z

Direct Variation – y varies directly as xy = kx

*Note: k = constant of variation (a #)Inverse Variation – y varies inversely as x

y = k x

Joint Variation – y varies jointly as x and zy = kxz

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20

a

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse

a

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x , direct

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x , direct, k = -0.5

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x , direct, k = -0.5

c. A = ½bh

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x , direct, k = -0.5

c. A = ½bh , joint

Ex. 1 State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.

a. ab = 20

b = 20 , inverse, k = 20

a

b. y = -0.5

x

y = -0.5x , direct, k = -0.5

c. A = ½bh , joint , k = ½

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

18 = k(15)

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

18 = k(15)

18 = 15k

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

18 = k(15)

18 = 15k

18 = k

15

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx

18 = k(15)

18 = 15k

18 = k

15

6 = k

5

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx 2) y = kx

18 = k(15)

18 = 15k

18 = k

15

6 = k

5

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx 2) y = kx

18 = k(15) y = 6x

18 = 15k 5

18 = k

15

6 = k

5

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx 2) y = kx

18 = k(15) y = 6x

18 = 15k 5

18 = k

15

6 = k

5

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx 2) y = kx

18 = k(15) y = 6x

18 = 15k 5

18 = k y = 6(20)

15 5

6 = k

5

Ex. 2 Find each value.

a. If y varies directly as x and y = 18 when x = 15, find y when x = 20.

1) y = kx 2) y = kx

18 = k(15) y = 6x

18 = 15k 5

18 = k y = 6(20)

15 5

6 = k y = 120

5 5

Ex. 2 Find each value.a. If y varies directly as x and y = 18

when x = 15, find y when x = 20.1) y = kx 2) y = kx

18 = k(15) y = 6x 18 = 15k 5

18 = k y = 6(20) 15 5

6 = k y = 120 5 5

y = 24

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

y = 1xz

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

y = 1xz

y = 1(9)(-5)

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

y = 1xz

y = 1(9)(-5)

y = -45

c. If y varies inversely as x and y = -14 when x = 12, find x when y = 21

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

y = 1xz

y = 1(9)(-5)

y = -45

c. If y varies inversely as x and y = -14 when x = 12, find x when y = 21

1) y = k

x

-14 = k

12

-168 = k

b. Suppose y varies jointly as x and z. Find y when x = 9 and z = -5, if y = -90 when z = 15 and x = -6

1) y = kxz

-90 = -6(15)k

-90 = -90k

1 = k

2) y = kxz

y = 1xz

y = 1(9)(-5)

y = -45

c. If y varies inversely as x and y = -14 when x = 12, find x when y = 21

1) y = k

x

-14 = k

12

-168 = k

2) y = k

x

21 = -168

x

x = -8