1.3 Rates of Change in Linear and Quadratic Functions

Welcome to Topic 1.3: Rates of Change in Linear and Quadratic Functions!

In the previous lessons, we explored how two quantities change together and how to calculate the average rate of change. Now, we are going to look at two specific "superstars" of the function world: Linear Functions and Quadratic Functions. By the end of these notes, you will be able to identify these functions just by looking at how their rates of change behave.

Note: If you need a refresher on the basic formula for average rate of change, check out Topic 1.2: Rates of Change.

1. Linear Functions: The Constant Climbers

Linear functions are the simplest functions to understand because they are perfectly consistent. Whether you look at the beginning, the middle, or the end of the function, the "steepness" never changes.

Key Characteristic: Constant Rate of Change

For a linear function, the rate of change is constant. This means that for every equal increase in the input \( x \), the output \( y \) changes by the exact same amount. This constant rate is what we usually call the slope.

The Rule: If a function is linear, the average rate of change between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is always the same value, \( m \).

\( \text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = m \)

Visualizing Linear Change

Imagine you are walking up a ramp. If every step you take forward moves you exactly 2 inches up, you are moving at a constant rate. In a table of values, you can spot this easily:

\( x \): 0, 1, 2, 3
\( y \): 5, 8, 11, 14

Notice that as \( x \) increases by 1, \( y \) always increases by 3. Because this "first difference" is constant, the function is linear.

Quick Takeaway: If the rate of change is a constant number (like 5, -2, or 0), the function is linear.

2. Quadratic Functions: The Changing Climbers

Quadratic functions are a bit more "dynamic" than linear functions. Their rate of change is not constant—it is always changing. However, there is a very specific pattern to how it changes.

Key Characteristic: Linear Rate of Change

For a quadratic function, the rate of change itself changes at a constant rate. This means that if you calculate the average rates of change over consecutive equal-length intervals, those values will form a linear pattern.

The Rule: For a quadratic function, the average rate of change is a linear function of \( x \).

The "Second Difference" Trick

If you have a table where the \( x \)-values increase by the same amount, you can identify a quadratic function by looking at the "change in the change":

\( x \): 0, 1, 2, 3, 4
\( y \): 0, 1, 4, 9, 16

1. Find the First Differences (Rate of Change):
\( 1-0 = 1 \)
\( 4-1 = 3 \)
\( 9-4 = 5 \)
\( 16-9 = 7 \)
(These are 1, 3, 5, 7... Not constant! So it's not linear.)

2. Find the Second Differences (Change in the Rate of Change):
\( 3-1 = 2 \)
\( 5-3 = 2 \)
\( 7-5 = 2 \)
(These are 2, 2, 2... Constant!)

Because the second differences are constant, we know the original function is quadratic.

Quick Takeaway: If the rate of change is changing at a constant rate (linear), the function is quadratic.

3. Comparing Linear and Quadratic Rates

To keep these straight, use this simple mental hierarchy:

• Linear Functions: The output changes at a constant rate.
• Quadratic Functions: The output changes at a linear rate.

Did you know? This pattern continues! If the rate of change were quadratic, the function would be cubic (a polynomial of degree 3). But for this chapter, we focus specifically on the relationship between linear and quadratic behaviors.

4. Common Pitfalls to Avoid

Mistake 1: Confusing "Rate of Change" with "Function Value"
Just because a function's values are increasing doesn't mean the rate of change is constant. Always subtract the outputs to check the rate at which they are growing.

Mistake 2: Forgetting the Interval Length
When checking for constant or linear rates of change in a table, ensure your \( \Delta x \) (the change in input) is the same for every step. If the \( x \)-values skip around, you must divide \( \Delta y \) by \( \Delta x \) for each interval to find the actual average rate of change before comparing them.

Mistake 3: Misinterpreting Negative Rates
A constant negative rate of change (like -4) still means the function is linear. It just means the line is going down instead of up!

5. Summary Checklist

When you encounter a function representation (table, graph, or equation), ask yourself:

1. Is the average rate of change the same for any two points? If yes \( \implies \) Linear.

2. Is the average rate of change changing at a constant rate over equal-length intervals? If yes \( \implies \) Quadratic.

3. Is the graph a straight line? If yes \( \implies \) Constant Rate of Change (Linear).

4. Is the graph a parabola? If yes \( \implies \) Linear Rate of Change (Quadratic).

Next Step: We will build on this in Topic 1.4 by looking at how these ideas apply to higher-degree Polynomial Functions!