Slope. Direct Variation

What you’ll learn

To write and graph an equation of a direct variation.

Direct variation and Constant of variation for a direct variation

To find the slope using a graph without the formula.

Problem 1: Finding the slope.If you travel at 50 mph for one hour, then you would have traveled 50 miles. If you travel for 2 hours at that speed, you would have traveled 100 miles. 3 hours would be 150 miles, etc. To graph these you could use a table

Time (h)

Distance

Ordered pair

1 50 (1,50)

2 100 (2,100)

3 150 (3,150)

4 200 (4,200)

Let’s be y= the distanceAnd x=the time(hours)

The graphs of the ordered pairs(time ,distance) lie on a line. The relationship between time and distance is linear. When data are linear; the rate is constant.

0 1 2 3 4 5 6

300

50

100

150

200

250

Notice that the rate is just ratio of the rise over the run between two points. This rate of change is called slope

runrise

change horizontalchange vertical

slope

122

144

133

166

There is a worksheet for finding slope from the the graph only. KUTA and you have to do it now

Hours Worked (x)

1 2 3 4

Dollars Earned (y)

10 20 30 40

If John earns $10.00 an hour, the above table showsthe relationship between hours worked and dollars earned. From this table, we can see that dollars earned is equal to hours worked multiplied by 10.

The equation y=10x describes this relationship. In this relationship, the number of dollars earned varies directly with the number of hours worked.

Problem 2: Direct Variation

A direct variation is a relationship that can be represented by a function in the Form y=kx, where k‡0. the constant of variation for a direct variation k is the coefficientof x.

By dividing each side by x you can see the ratio is constant

kx

yx

kx

x

y

kxy

The money collected and the hours worked

The distance traveled and the speed of the car

The circumference of a circle and its diameter

The area of a circle and its radius

M=k•h

d=k•t

C=k•d

2rkA

Examples of direct variation

Problem 1: Does the equation represent a direct variation?Identifying a direct variation

a) x2y7

7x2

7y7

x72

y

b)

x48y3

x34

38

y

First:Solve the equation for y

The equation has the Form Y=kx, so the equation is a direct variation. Its constantof variation is

72

You cannot write the equation in the formy=kx. It is not a directvariation

8x4y3

Your turn

Does 4x+5y=0 represent a direct variation, if soFind the constant of variation.

54

k

x54

y

Answer:

Graphing a direct variation

Weight on Mars y varies directly with weight on Earth x . The weights of the Science instruments onboard the Phoenix Mars Lander on Earth and mars are shown.

Weight on Mars 50 lbs and the weight on Earth 130lbs

a) What is an equation that relates weight, in poundson Earth x and weight on Mars y?

kxy )130(k50

k38.0 x38.0y

What is the graph?

x 0.38x y0 0.38(0) 0

50 0.38(50) 19100 0.38(100) 38150 0.38(150) 57

Ordered pairs (x,y)(0,0)(50,19)(100,38)(150,57)

50 100 150

60

40

20

Your turn

Weight on the moon y varies directly with the weight on Earth x. A person who weights 100lb on Earth weights 16.6 lb on the moon. What is an equation that relates weight on Earth x and weight on the moon y? what is the graph of this equation.

x166.0y 2 4 6

0.4

0.2

Answer:

Take a note: the graph of a direct variationequation y=kx is a line with the followingproperties.•The line passes through (0,0)•The slope of the line is k

When k›0 when k‹0

xx

yy

Writing a Direct Variation From a Table

For the data in the table, does y vary directlywith x? if it does, write an equation for the direct variation.

x y

4 6

8 12

10 15

x y

-2 3.2

1 2.4

4 1.6

a) b)

Find x

y for each ordered pair

5.11015

812

46

4.2

14.2

6.12

3

Y=1.5x So y does not vary directly

Your turn

For the data in the table, does y vary directlywith x? if it does, write an equation for the direct variation.

x y

-3 2.25

1 -0.75

4 -3

x y

3 5.4

7 12.6

12 21.6

x y

-6 9

1 -1.5

8 -12

A B C

xyyes 75.0, x8.1y,yes

x5.1y,yes Answers