9.1 Inverse & Joint 9.1 Inverse & Joint VariationVariation

By: L. Keali’i AliceaBy: L. Keali’i Alicea

Just a reminder from chapter 2Just a reminder from chapter 2

Direct Variation

Use y=kx.

Means “y varies directlyvaries directly with x.”

k is called the constant of variationconstant of variation.

New stuff!New stuff!

Inverse VariationInverse Variation

“y varies inverselyvaries inversely with x.”

k is the constant of variationconstant of variation.

x

ky

Ex: tell whether x & y show direct variation, Ex: tell whether x & y show direct variation, inverse variation, or neither.inverse variation, or neither.

a. xy=5

b. x+y = 7

c. 5.1

yx

Hint: Solve the equation for y and take notice of

the relationship.

xy

5

Inverse Variation

Neither

xy 5.1

Direct Variation

Ex: The variables x & y vary inversely. Use Ex: The variables x & y vary inversely. Use the given values to write an equation relating the given values to write an equation relating

x and y. Then find y when x =4. x and y. Then find y when x =4. • x=2, y=4

k=8• Find y when x= 4.

y= 2

x

kyuse :

24

k

x

ky

xy

8

4

8y

Ex: The variables x & y vary inversely. Use Ex: The variables x & y vary inversely. Use the given values to write an equation relating the given values to write an equation relating

x and y. Then find y when x =4. x and y. Then find y when x =4. • x=16, y= 1/4

k=(1/4)16= 4• Find y when x= 4.

y= 1

x

kyuse :

164/1

k

x

ky

xy

4

4

4y

Joint VariationJoint Variation

• When a quantity varies directly as the product of 2 or more other quantities.

• For example: if z varies jointly with x & y, then z=kxy.

• Ex: if y varies inversely with the square of x, then y=k/x2.

• Ex: if z varies directly with y and inversely with x, then z=ky/x.

Examples: Write an equation.Examples: Write an equation.

• y varies directly with x and inversely with z2.

• y varies inversely with x3.

• y varies directly with x2 and inversely with z.

• z varies jointly with x2 and y.

• y varies inversely with x and z.

2z

kxy

3x

ky

z

kxy

2

ykxz 2

xz

ky

The variable z varies jointly with x and y. Use the The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and given values to write an equation relating x, y, and

z. Then find z when x= -3 and y= 4.z. Then find z when x= -3 and y= 4.

• x= 1, y=2, z=6We use:z = kxyWe put in the values for z, x, and y to solve for k.6= k(1)(2) Then solve for k.k= 6/2 = 3z = 3xyWe then put in the values for x and y and solve for

z.

z= 3(-3)(4)= -36

AssignmentAssignment

9.1 B (2-12 even, 13-18)