Algebra ( Period 5)

Chapter 3-4 Direct Variation

We’ve learned that the unit rate is the constant rate of change in a linear relationship and that it’s the slope of a line when it’s graphed. We’ve also learned that if a graph of an equation goes through the origin (0,0)  it’s proportional  and the ratio of any y value to it’s x value is a constant (which turns out to be the unit rate or constant rate of change or slope of the line)

When the linear relationship is proportional, we say it’s a DIRECT VARIATION. Now the constant rate of change, the slope, the unit rate, is called the CONSTANT OF VARIATION or the CONSTANT OF PROPORTIONALITY

This is not a new concept. IT IS  just NEW VOCAB!

We also say: y varies directly (constantly) with x.

The slope is now replaced by the letter k instead of m

Finding the equation of a line that is proportional

Find k (the slope) by counting the rise/run of the graph

Write the equation using the format  y = kx

Notice: if you always pick the origin as the point to count rise/run from—the slope (k) is always just y/x

In a word problem, if it says one amount VARIES DIRECTLY with another, you know that the origin is one of the points!!

You also know that the equation is y = kx 
YOU just need to find k
and k is y/x of any point OTHER THAN THE ORIGIN

A babysitting example

The amount of money earned  VARIES DIRECTLY with the time worked.

THINK: the graph and equation go through (0,0)

THINK: Any other point will give you the slope, or constant of proportionality, or unit rate ( all the same thing) SO you only need one additional point.

We are given that she earns $30 for 4 hours. Find the equation.

Rise/Run = y/x
BECAUSE THEY SAID IT VARIED DIRECTLY!!

k = 30/4
Simplify
k = 7.5
So the equation is y = 7.5x

What does the 7.5 represent?
The unit rate of $7.50/ hour of babysitting!

A bicycling example 
The distance the cyclist bikes in miles VARIES DIRECTLY with the time in hours that he bikes.

THINK: The graph and equation go through the origin (0,0).THINK: Any other point will give you the slope, or constant of proportionality, or unit rate (all the same thing) SO you only need one additional point.

He bikes 3 miles in ¼ hour. Find the equation.

Rise/run = y/x
BECAUSE THEY SAID IT VARIES DIRECTLY

k = 3/¼  or 3/.25 Now the hardest part is doing this 3/.25

If you kept it as 3/¼  you could read this as 3 divided by ¼
THINK: instead of dividing, multiply by the reciprocal of ¼

or 3 (4/1) = 12 (Wait, wasn’t that much easier than dividing 3 by .25!!
k = 12

The equation is y = 12x

What does the 12 represent?

The unit rate of 12 miles/ hour – that’s the cyclist’s speed 12mph

  Determining whether a Table of Values is Direct Variation If you are given a table of values, you can determine if the relationship is direct variation by dividing 3 y’s by their x values and making sure that you get the SAME value. If you do, it is proportional, goes through the origin (0,0) and the slope of y/x is the unit rate ( which is now called the constant of variation)!

Example

Given 3 points (5, 20) , (6, 24), and (7, 28):

Divide each y/x

20/5 = 4

24/6 = 4

28/7 = 4

Since all the ratios simplify to the same value (4), it is a direct variation. The slope of 4 is the unit rate, which is the constant rate of change and is now also called the constant of variation.

Finding Additional Values for the Direct Variation once you have the Equation
Once you have the equation y = kx, you can find infinite additional values (points) that will work.
For example, in the first babysitting example, the equation is y = $7.50x, which we write as y = 7.5x  If she babysits for 20 hours, how much did she earn?
x = 20

so y = 7.5(20) = 150 so She earns $150.
If she earns $750, how many hours did she need to work?

Now y = 750  so  750 = 7.5x 
It is a one-step equation and we get
x = 100 or 100 hours!

 Finding the Equation if you know 1 point and then Finding Additional Values

y varies directly with x. Write an equation for the direct variation. Then find each value

If y = 8 when x = 3, find y when x = 45

FIRST you need to find k

y = kx… In this case we have 8 = k(3) or 8 = 3k

Solve this 1 step equation—leaving it in fraction form!

8/3= k

so

y = (8/3)x

Now, find y when x = 45

y = (8/3)(45)

solve

y = 120

Applying direct variation to the Distance Formula d = rt

A jet’s distance varies directly as the hours it flies

If it traveled 3420 miles in 6 hours, how long will it take to fly 6500 miles?

k = 3420/6 = 570mph ( its speed)

6500 = 570t

t ≈11.4

about 11.4 hours