Direct Variation Rate of Change

Direct Variation

Direction variation is used in many every day life applications. One example is distance; d=rt is just one example of a direction variation.

Before going further... take notes of this lesson.

Remember the teacher reviews the final scores and can adjust on her end if necessary.

Direct Variation defined

• A direct variation is a relationship between two variables, commonly x and y, that is constant.

• A direct variation can be written in the format y = kx, k  $\ne$  0

• A direct variation resembles          y = mx+b with b ALWAYS = 0! so we don't write it.

Constant of Variation

In the formula, y=kx, k is referred to as the constant of variation OR the constant of proportionality. It is the coefficient of x (not y!)

A constant does not change in value.

What does it take to be a direct variation?

• The line on a graph will ALWAYS pass through the origin!

• Nothing is added or subtracted from kx in the formula; there is no "b". It is simply y = kx (y = mx would also work)

• All ordered pairs are proportional. If you divide the y from one ordered pair by its x and do that for all ordered pairs, the ratios will be equal!

Multiple Select

Which of the following ARE direct variations based on the definition and the three requirements?

• 3x + 5y = 1

• -12x = 6y

Fill in the Blank

2x + y = 0 is an example of a direct variation and the constant of variation is _____ (do the math!)

Multiple Select

Is -x +y = 1 an example of a direct variation?

• Yes

• No

How to graph a DIRECT VARIATION

Step 1. Plot a point on the origin, (0,0).

Step 2. Use k like you would slope to plot two more points.

Step 3. Connect the points with a line (with arrows on each end!)

Writing a DV from a graph

1. Using y = kx and solve for k by dividing both sides by x:   $\frac{y}{x}$  =k
2. Substitute in the x and y-values of the point on the direct variation to determine the constant of variation:    $\frac{2}{-1}$  =k or k = -2

3. Rewrite the DV equation with only the value of k: y = -2x

Fill in the Blank

The direct variation equation for the real life example shown on the graph is ........

Y varies directly with x

This video is less than two minutes in length and shows one way of solving "varies directly with x" problems. ("directly proportional problems are done the same way)

Open Ended

Why does C vary directly with s?

Open Ended

What is the direct variation equation that relates C to s?

How to determine a direct variation from a table

Remember that y = kx is the same as y = ax.

Constant of proportionality from tables

For each ordered pair given, (-4,1),(-3,0.75). (-2. 0.5) and (-1,0.25) if you divide the y value by the x value you find k for the ordered pairs. IF k is the same for each, then you have a direct variation and have found the constant of variation/the constant of proportionality

Multiple Select

Does the table illustrate a direct variation?

• No

• Yes

How did you do?

If you watched this lesson and got everything right THE FIRST TIME you will get a grade of 100. If you missed anything you will get a grade no lower than 75... If you rewatch it to try to improve your grade you can get up to an 88.

Hopefully you took notes as you were watching!

You will have a practice sheet to work on that goes with this as well.