Algebra 1 > Notes > Yorkcounty Final > Yc > Lesson 23

Lesson 23 Direct and Inverse Variation

Direct Variation • The relationship that exists between two variables when a change in one variable causes a proportional change in the same direction in the other variable. • y = kx

( k is a constant)

• Simply said: Direct Variation – as x gets bigger y gets bigger

Direct Variation

Direct Variation • y = kx • This is a linear function with the graph passing through the origin. • y = mx + b ; m is the slope b is the y-intercept y = kx ; k is the slope (which is a constant) and the y- intercept is zero.

From the information in the example, we can pull out some basic properties of direct variations: 1) y/x is always the same value this value “k” is called the constant of variation So, y/x = k 2) the general equation of a direct variation is y = kx 3) this equation is linear--remember y=mx + b If y = kx, then k is the slope and the y-intercept is zero. Hence, every direct variation is a graph of a line through the origin (0,0)

From the table below, determine if y varies directly as x. If so, write the variation algebraically.

X

-2

-1

3

5

Y

6

3

-9

-15

A) not a direct variation B) direct variation: y = -3x C) direct variation: y =-4x D) direct variation: y = 3x

To check to see if it is a direct variation, we will check to see if y/x is a constant value--the same every time: 6/-2

=

-3

3/-1

=

-3

-9/3

=

-3

-15/5 =

-3

Since we got the same value for all data points, the variation is direct. The constant of variation is -3. Thus the equation of variation is y = -3x. The best answer choice is B)

Try These: From the tables below, determine if y varies directly as x. If so, write the equation of variation.

1)

2)

x

3

-2

5

6

y

12

-8

20

12

x

-4

-6

12

-2

y

6

9

-18

3

Try This Solution: 1) Check to see if y/x is constant

2)

12/3

=

4

-8/-2

=

4

20/5

=

4

12/6

=

2

6/-4

=

-1.5

9/-6

=

-1.5

-18/12 =

-1.5

3/-2

-1.5

=

No, this is not direct because we did not get the same value for all y/x in the table.

This table is a direct variation. The equation of variation is y= -1.5x.

Y varies directly as x. Y is 24 when x is 3. Find y when x is 4.

Solution: Let us start by setting up a table: x

24

4

y

3

?

We must find y when x is 4: Since we are told that the relationship is a direct variation we can find the constant of variation by y/x. \ 24/3 = 8

So the variation equation is y=8x.

If we substitute 4 in for the x in our variation equation we get the answer: y = 8x y = 8(4) y = 32

Real World Problem • If it takes 6 hours to travel 300 miles, how many hours will it take to travel 900 miles? Assume the speed remains constant. Time is directly proportional to distance traveled. • T = kD ( time is directly proportional to distance traveled) • First solve for k 6 = k300 k = .02 T = .02D T = .02(900) T = 18 hours

At a given time and place, the height of an object varies directly as the length of its shadow. If a flagpole 6m high casts a shadow of 10m long, find the height of a building that casts a shadow 45 m long.

Again, we will make a table to represent our data: x: object ht.

6m

?

y: Shadow length

10m

45m

Find the constant of variation y/x = k

10/6 = 1.6

Find the equation of variation y = 1.6 x Use the 45m to find the height of the building y = 1.6x 45 = 1.6x 28.125 = x

divide both sides by 1.6

Inverse Variation

Consider a situation in which you are making rectangles of different sizes that have the same area which is 48. If x represents the length of the rectangle and y represents the width, you get a table of this type: x

y

1

48

2

24

3

16

4

12

6

8

We would say that the width of the rectangle is related inversely to the length. As one increases, the other decreases.

Inverse Variation

If we take another look at the table we will x notice that every time we multiply the y1 value by its corresponding x-value, we get the same number-- 48 2

y

(1)(48) = (2)(24) = (3)(16) = (4)(12)=

3

16

4

12

6

8

(6)(8)= 48 This value of 48 is called the “constant of variation”. We could make an equation: 48 = xy

or

y= 48 x

48 24

From the information in the example, we can pull out some basic properties of inverse variations: 1) xy is always the same value this value “k” is called the constant of variation So, xy - k 2) the general equation of a inverse variation is y = k/x

From the table below, determine if y varies inversely as x. If so, write the variation algebraically.

X

3

6

-12

-24

Y

8

4

-2

-1

To check to see if it is an inverse variation, we will check to see if xy is a constant value--the same every time:

3(8)

=

24

6(4)

=

24

-12(-2) =

24

-24(-1) =

24

Since we got the same value for all data points, the variation is inverse. The constant of variation is 24. Thus the equation of variation is xy = 24 or y = 24/x

Check to see if this is an Inverse Variation. x

2

4

-10

-24

y

10

5

-2

-4

Check to see if xy is constant 2(10) =

20

4(5)

=

20

-10(-2) =

20

-24(-4) =

96

Solution

No, this is not inverse because we did not get the same value for all xy in the table.

Y varies inversely as x. Y is 4 when x is 22. Find x when y is -11

Solution: Let us start by setting up a table: x

22

?

y

4

-11

We must find x when y is -11: Since we are told that the relationship is an inverse variation we can find the constant of variation by xy. 4(22) = 88

So the variation equation is xy=88.

If we substitute -11 in for the y in our variation equation we get the answer: xy

=

88

x(-11)

=

88

x(-11)/(-11)=

88/(-11)

x = -8

Real World Problem • The time is takes for a crew to paint a room varies inversely with people. • If it takes 3 people 5 hours to paint a room, how long will it take 5 people?

Solution • • • • • • • •

Time varies inversely with people. T = k/P First solve for k It took 3 people 5 hours to paint. 5 = k/3 k = 15 T = 15/P How long will it take 5 men? T = 15/5 Time = 3 hours.

• It wi ll take 5 hours for 3 men and 3 hours for 5 men.

The volume of gas varies inversely as the pressure. If the volume is 80m3 under 4 kg of pressure, find the volume under 10kg of pressure

Solution: Since we were told this is an inverse variation we can set up the table:

x

pressure

4kg

y

volume

80m3

10kg ?

To find the constant of variation we multiply the y by the x:

xy (4)(80) 320

So the equation of the variation is:

y = 320/x

Now we can use the equation to solve for our missing variable by substituting the x-value of 7 y = 320 / x y = 320/ 10 y = 32m3