Lesson 5.5

Direct Variation

If you double the recipe, you double the flour. If you work twice as long, you get paid twice as much. This perfect sync is called Direct Variation.

Introduction

Some lines are special. They start exactly at zero. Because of this, the ratio between input and output is always the same. We call this relationship Direct Variation or being Proportional.

Past Knowledge

Lesson 5.4. You know about Rate of Change. Today, the rate is "constant" in a very specific way.

Today's Goal

Identify functions where and find the Constant .

Future Success

This prepares you for "Slope-Intercept Form" where graphs DON'T start at zero ().

Key Concepts

The Constant of Proportionality ()

It's just the Slope, but simpler.

Equation

No allowed. Must pass through origin.

Finding k

Divide by anywhere on the table.

is not on the line, it is NOT Direct Variation.

Worked Examples

Example 1: Is it Direct?

Basic

x	y
2	6
4	12
5	15

Test using .

Row 1

Row 2

Conclusion

Since the ratio is constant (), Yes. Equation: .

Example 2: The Graph Test

Intermediate

Does this graph show direct variation?

Check Origin

Does it pass through ?

No. It passes through .

Not Direct Variation

Even though it is a line, it is not proportional because it doesn't start at zero.

Example 3: Solve for y

Advanced

If varies directly as , and when , find when .

Step 1: Find k

So the rule is .

Step 2: Use k

Plug in to the rule.

Common Pitfalls

Adding Constants

If you see , it is NOT direct variation. The ruins the proportion. It must be strictly multiplication.

Flipping k

is always . Think "Price per Item" (Dollars / Apples). Usually is on top.

Real-Life Applications

Currency Exchange: If 1 USD = 0.90 Euro, this is direct variation. . If you have 0 Dollars, you have 0 Euros. If you have 100 Dollars, you have 90 Euros. The constant is the exchange rate.