Linear and Nonlinear Functions

Lesson 3 of 3: Functions

In this lesson:

• Why is a linear function
• Spotting nonlinear functions and classifying any function

Learning Objectives: Linear and Nonlinear Functions

By the end of this lesson, you should be able to:

1. Explain why defines a function
2. Identify a linear function by its constant rate of change
3. Explain why a linear graph is a straight line
4. Give examples of functions that are not linear
5. Classify a function as linear or nonlinear and justify it

Same Shape — or Not?

A streaming service: $5 a month plus $3 per movie.

A growing square: area equals side times side.

Plot the cost as movies grow, and the area as the side grows. Same shape?

Building the Change in y Column

For , the output goes up by 3 every single step.

A Constant Rate Means Linear

When the rate of change is constant, the function is called linear.

For , the rate is always — so it's linear.

Constant step is the whole idea.

Why the Graph Is a Straight Line

• Equal steps right, equal steps up
• The points land on a straight line

A Line Can Go Down

For , the change in is every step.

Still constant — so still linear — just sloping down.

Predict: Does Decreasing Mean Nonlinear?

goes down as grows.

Is a decreasing function nonlinear? Predict yes or no before advancing.

Horizontal Lines Count as Linear Too

gives output for every input.

The change in is — and is constant.

So a horizontal line is linear, rate of change .

Quick Check: Is the Rate Constant?

Table A:

Table B:

For each, is the rate of change constant?

What If the Step Isn't Constant?

So far, equal steps. But what about functions where the output speeds up or slows down?

Those are coming next — and their graphs won't be straight.

A = s² Is Not Linear

The differences keep growing — not constant.

A = s² Curves Upward

• The points bend upward, not a straight line
• Growing steps a curve

Another Curve: y = 2ˣ

Differences double — definitely not constant.

Linear vs. Nonlinear, Side by Side

• Left: — constant — linear
• Right: — growing — nonlinear

Create Your Own Nonlinear Function

Invent a rule where the rate of change is not constant.

Build a 5-row table, fill the change column, and show it varies.

Try , , or one of your own.

One Property, Now a Third Test

The table test (constant step) and the graph shape (line vs. curve) check the same thing.

Now add a third test: the equation itself.

Classify Straight From the Equation

Linear if is to the first power only (or absent, like ).

The Table Trap: Check the x-Steps

Divide: . Constant — linear!

Two Separate Questions, Not One

"Nonlinear" does not mean "not a function."

1. First: is it a function? (each input, one output)
2. Then: is it linear? (constant rate of change)

is a function — and nonlinear.

Your Turn: Classify on Your Own

On your whiteboard, classify by the power of :

Write linear or nonlinear for each. No calling out.

Classify All Three Functions Now

Classify and justify each by constant rate of change:

1. Equation:
2. Table:
3. Graph: a straight line sloping down

Always Checking the Same One Thing

✓ Linear = constant rate of change = straight line

✓ A line can go up, down, or flat; check steps, not direction

✓ Unequal -steps? Divide. Nonlinear not a function

Next: in high school, this split grows into a whole family of functions.