slope. direct variation what you’ll learn to write and graph an equation of a direct variation....
TRANSCRIPT
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
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