Rates of Change in Linear and Quadratic Functions
Why this matters
Imagine you're on a cross-country road trip from Boston to Seattle. For a long stretch through the plains of Montana, you set your cruise control to 70 mph. Your speed is constant. This is a linear relationship: for every hour that passes, you cover 70 miles. The rate of change is unchanging.
Now, picture yourself exiting the highway in Chicago. You slow down for the off-ramp, stop at a light, and then accelerate back onto a local road. Your speed is constantly changing—it's not a steady 70 mph anymore. This is more like a quadratic function. Your rate of change (your speed) is itself changing over time.
In this lesson, we'll learn how to precisely measure that "average speed" for any function, which we call the average rate of change. We'll see why it's simple for linear functions and uncover a surprising, predictable pattern for quadratic functions that will be essential for your AP exam.
Concept overview
flowchart TD
A[Start with a function and equal-length intervals] --> B{Calculate Average Rates of Change (AROCs) for each interval};
B --> C{Are the AROCs constant?};
C -->|Yes| D[Function is LINEAR];
D --> E[Change in AROCs is 0];
C -->|No| F{Calculate the differences between consecutive AROCs};
F --> G{Are these new differences constant?};
G -->|Yes| H[Function is QUADRATIC];
H --> I[Change in AROCs is a non-zero constant];
G -->|No| J[Function is another type];
Core explanation
Hello everyone, it's Saavi. Today we're diving into one of the most fundamental ideas in all of calculus: rates of change. It sounds fancy, but you've been working with it for years.
What is a Rate of Change?
Anytime you've calculated the slope of a line, you've found a rate of change. It's the ratio that tells you how much the vertical value (y) changes for every one-unit change in the horizontal value (x).
You know the slope formula: m = (y₂ - y₁) / (x₂ - x₁)
This is the blueprint for everything we'll do today.
Average Rate of Change (AROC) and the Secant Line
For a straight line, the slope is the same everywhere. But what about a curve, like a parabola? The "steepness" is constantly changing.
We can't find the slope at a single point yet (that's a calculus topic!), but we can find the average rate of change between two points.
Imagine picking two points on a curve, (a, f(a)) and (b, f(b)), and drawing a straight line through them. This line is called a secant line. The slope of that secant line is the average rate of change of the function between x=a and x=b.
The formula is the same as the slope formula, just with function notation:
Average Rate of Change (AROC) = (f(b) - f(a)) / (b - a)
Memorize this. It's non-negotiable. It represents the average "speed" of the function across the interval [a, b].
Rates of Change for Linear Functions
Let's start with something familiar: a linear function like f(x) = 3x + 5.
Let's find the AROC on the interval [1, 4].
Here, a=1 and b=4.
f(1) = 3(1) + 5 = 8f(4) = 3(4) + 5 = 17
AROC = (f(4) - f(1)) / (4 - 1) = (17 - 8) / 3 = 9 / 3 = 3.
The AROC is 3. Notice anything? It's the slope of the line. For any linear function, the average rate of change over any interval will always be the same constant value: its slope. Because the rate of change is constant, the change in the rate of change is always zero.
The Big Idea: Rates of Change for Quadratic Functions
This is where it gets interesting. Let's analyze the simple quadratic function f(x) = x². We're going to look at the AROC over several consecutive intervals of equal length. Let's use intervals of length 1.
- Interval [0, 1]AROC =
(f(1) - f(0)) / (1 - 0) = (1² - 0²) / 1 = 1 - Interval [1, 2]AROC =
(f(2) - f(1)) / (2 - 1) = (2² - 1²) / 1 = (4 - 1) / 1 = 3 - Interval [2, 3]AROC =
(f(3) - f(2)) / (3 - 2) = (3² - 2²) / 1 = (9 - 4) / 1 = 5 - Interval [3, 4]AROC =
(f(4) - f(3)) / (4 - 3) = (4² - 3²) / 1 = (16 - 9) / 1 = 7
Look at our average rates of change: 1, 3, 5, 7. They aren't constant! They are increasing. This tells us the graph is getting steeper.
Now, let's take the next step. Let's look at the change in the average rates of change.
- From 1 to 3, the change is
+2. - From 3 to 5, the change is
+2. - From 5 to 7, the change is
+2.
The change is constant!
This is the big reveal for this topic:
- For a linear function, the rate of change is constant.
- For a quadratic function, the rate of change is not constant, but the change in the rate of change is constant over equal-length intervals.
This is how you can identify a quadratic function from a table of data. If the first differences are not constant but the second differences are, the data likely models a quadratic function.
AROC and Concavity
This pattern has a name in the visual language of graphs: concavity.
-
When the average rates of change are increasing (like
1, 3, 5, 7), the function's slope is getting steeper. The graph bends upward, like a bowl. We call this concave up. -
If the average rates of change were decreasing (e.g.,
7, 5, 3, 1), the function's slope would be getting less steep. The graph would bend downward, like a dome. We call this concave down. This happens with parabolas that open downward, likef(x) = -x².
So, by analyzing how the AROC changes, we can describe the curvature of the graph itself.
Worked examples
Let's walk through a couple of problems you might see on a test. The key is to be slow, methodical, and write out each step.
Problem 1: Calculating AROC for a Quadratic
Find the average rate of change of the function
g(x) = 2x² - 3x + 1on the interval[-1, 3].
Solution:
- 1Identify the formula and your inputsThe formula is
AROC = (g(b) - g(a)) / (b - a). Here,a = -1andb = 3. - 2
Calculate
g(a)andg(b)separately. This is where most students make calculation errors. Take your time.g(b) = g(3) = 2(3)² - 3(3) + 1 = 2(9) - 9 + 1 = 18 - 9 + 1 = 10g(a) = g(-1) = 2(-1)² - 3(-1) + 1 = 2(1) + 3 + 1 = 2 + 3 + 1 = 6
- 3Plug the values into the AROC formula
AROC = (10 - 6) / (3 - (-1)) - 4Simplify carefullyWatch your signs in the denominator!
AROC = 4 / (3 + 1) = 4 / 4 = 1
Problem 2: Using a Table to Identify a Function Type
A function
h(x)is defined by the table below. Ish(x)linear or quadratic?
x h(x) 0 5 2 11 4 21 6 35 8 53
Solution:
-
Check for a constant rate of change (Linear). The
xvalues are increasing by a constant amount (+2), so we can compare the rates of change. We'll calculate the AROC for each interval.- Interval
[0, 2]:AROC = (11 - 5) / (2 - 0) = 6 / 2 = 3 - Interval
[2, 4]:AROC = (21 - 11) / (4 - 2) = 10 / 2 = 5 - Interval
[4, 6]:AROC = (35 - 21) / (6 - 4) = 14 / 2 = 7 - Interval
[6, 8]:AROC = (53 - 35) / (8 - 6) = 18 / 2 = 9
The average rates of change are
3, 5, 7, 9. They are not constant, so the function is not linear. - Interval
-
Check for a constant change in the rate of change (Quadratic). Now we look at the list of AROCs we just calculated:
3, 5, 7, 9. Let's find the difference between consecutive terms.5 - 3 = 27 - 5 = 29 - 7 = 2
The change is a constant
2.
Try it yourself
Time to get your hands dirty. Try these out, and remember to show your work.
Problem 1:
A baseball is thrown upwards from the top of a building. Its height in feet after t seconds is given by the function h(t) = -16t² + 48t + 64. What is the average rate of change of the height (the average velocity) of the ball between t=1 second and t=3 seconds?
Hint: Your a is 1 and your b is 3. What does a negative AROC mean in the context of the ball's flight?
Problem 2:
Consider the function f(x) represented by the values in the table. Calculate the constant rate of change of the average rates of change.
| x | f(x) |
|---|---|
| -2 | 22 |
| -1 | 10 |
| 0 | 4 |
| 1 | 4 |
| 2 | 10 |
Hint: First, calculate the AROC for each interval: [-2, -1], [-1, 0], [0, 1], and [1, 2]. Then, find the difference between those resulting rates.
Practice — 8 questions
In simple terms, this topic is about measuring how fast a function's values are changing—its "average speed"—and how that speed itself changes for straight lines (linear functions) and U-shaped curves (quadratic functions).
- 1.3.A: Determine the average rates of change for sequences and functions, including linear, quadratic, and other function types.
- 1.3.B: Determine the change in the average rates of change for linear, quadratic, and other function types.
- 1.3.A.1
- For a linear function, the average rate of change over any length input-value interval is constant.
- 1.3.A.2
- For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.
- 1.3.A.3
- The average rate of change over the closed interval [a, b] is the slope of the secant line from the point (a, f (a)) to (b, f (b)).
- 1.3.B.1
- For a linear function, since the average rates of change over consecutive equal-length input-value intervals can be given by a constant function, these average rates of change for a linear function are changing at a rate of zero.
- 1.3.B.2
- For a quadratic function, since the average rates of change over consecutive equal-length input-value intervals can be given by a linear function, these average rates of change for a quadratic function are changing at a constant rate.
- 1.3.B.3
- When the average rate of change over equal-length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up. When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals, the graph of the function is concave down.
flowchart TD
A[Start with a function and equal-length intervals] --> B{Calculate Average Rates of Change (AROCs) for each interval};
B --> C{Are the AROCs constant?};
C -->|Yes| D[Function is LINEAR];
D --> E[Change in AROCs is 0];
C -->|No| F{Calculate the differences between consecutive AROCs};
F --> G{Are these new differences constant?};
G -->|Yes| H[Function is QUADRATIC];
H --> I[Change in AROCs is a non-zero constant];
G -->|No| J[Function is another type];
Read what Saavi narrates
Hi everyone, it's Saavi from Shrutam. Let's talk about rates of change.
Imagine you're on a road trip from Boston to Seattle. For a long stretch, you set your cruise control to 70 miles per hour. Your speed is constant. This is a linear relationship... for every hour that passes, you cover 70 miles. The rate of change is unchanging.
But now, picture yourself exiting the highway in Chicago. You slow down, stop at a light, then accelerate back onto a road. Your speed is constantly changing. This is more like a quadratic function. Your rate of change, your speed, is itself changing.
Today, we're learning to measure that "average speed" for any function. We call it the average rate of change. For straight lines, it's constant. But for curves, we'll find a really cool, predictable pattern.
Let's try a problem you might see on the exam. Imagine you're given a table of x and y values, and asked if the function is linear or quadratic. Let's say the x-values are 0, 2, 4, 6 and the corresponding y-values are 5, 11, 21, 35.
First, we check if it's linear by finding the rate of change for each step. From x equals 0 to 2, the y-value changes by 6. So the rate is 6 divided by 2, which is 3.
Now for the next step, from x equals 2 to 4. The y-value goes from 11 to 21, a change of 10. The rate is 10 divided by 2, which is 5.
Right away, we know it's not linear, because the rates 3 and 5 are different.
So, let's check if it's quadratic. We look at our list of rates... 3, 5... let's find the next one. From x equals 4 to 6, y goes from 21 to 35, a change of 14. The rate is 14 divided by 2, which is 7.
Our rates of change are 3, 5, 7. Do you see the pattern? The rate itself is changing, but it's changing by a constant amount! From 3 to 5 is plus 2. From 5 to 7 is plus 2. Because the *change* in the rate of change is constant, we know the function is quadratic.
Now, a very common mistake I see every year is students forgetting to divide by the change in x. They'll just subtract the y-values and think that's the rate. But a rate always compares two quantities. It's rise *over* run. So always remember to divide by that `b minus a` in the denominator.
You're building a powerful toolkit for understanding functions. Keep practicing, be patient with yourself, and you will master this.
Function notation is precise. `f(b) - f(a)` means you find the output at `b`, find the output at `a`, and then subtract. `f(b - a)` means you subtract `a` from `b` *first* and then plug that single new number into the function. These are completely different operations.
Always calculate `f(a)` and `f(b)` as two separate steps before you do any subtraction.
`f(b) - f(a)` is just the total change in the `y`-value. A *rate* of change must compare the change in `y` to the change in `x`. Forgetting the denominator means you've only done half the problem.
Write the full formula `(f(b) - f(a)) / (b - a)` on your paper before you plug in any numbers.
These are two different layers of analysis. The AROC is the "speed" of the function. The *change* in the AROC is the "acceleration" of the function. An AP question might ask for one or the other, and they have different values.
In a table problem, create a new column for your AROC calculations. Then, if needed, create a *third* column to find the differences in that second column. Keep your work organized.
A simple sign error like `3 - (-1) = 2` will cascade through your entire calculation and give you the wrong answer.
When you substitute a negative number, use parentheses. Write `(3 - (-1))` to remind yourself that you are subtracting a negative, which is addition.