AP Precalculus: Rate of Change
A quadratic function f(x) = ax² + bx + c with a ≠ 0 does not have a constant rate of change, because a parabola has no straight pieces. But when inputs are equally spaced, the rates of change themselves change by a constant amount. If that constant is positive, the parabola is concave up (a > 0). If it is negative, it is concave down (a < 0). If the rates change by uneven amounts, the function is neither linear nor quadratic.
1. Relations, Functions, and Function Notation
A relation pairs inputs with outputs. It is a function only when every input is paired with exactly one output. Two different inputs may share an output, but one input can never have two. Graphically, this is the vertical line test: if any vertical line crosses a graph more than once, the graph is not a function. Function notation f(x) names the output for input x, so f(3) means “replace x with 3.” The input can also be an expression, such as f(a + 1).
Worked Example: Is {(1, 4), (2, 7), (3, 4), (5, 9)} a function?
The inputs are 1, 2, 3, 5, and none repeats. The output 4 appears twice, which is allowed.
Yes, it is a function.
1. Which set of ordered pairs represents a function?
Explanation: Only the third set gives each input one output. In the others, the input 2, 4, or 1 appears with two different outputs. Repeated outputs (all 6s) are fine.
2. If f(x) = 3x² − x + 4, what is f(−2)?
Explanation: Substitute x = −2: 3(−2)² − (−2) + 4 = 3(4) + 2 + 4 = 12 + 2 + 4 = 18.
3. If v(t) = t² − 3t, which expression equals v(a + 1)?
Explanation: Replace t with (a + 1): (a + 1)² − 3(a + 1) = a² + 2a + 1 − 3a − 3 = a² − a − 2.
2. Domain, Range, Zeros, and Intercepts
The domain is the set of allowed inputs, and the range is the set of outputs that actually occur. Algebraically, two rules find most domains: never divide by zero, and never take an even root of a negative number. A zero of a function is an input that makes the output 0 (an x-intercept). The y-intercept is f(0), and a function has at most one of them.
Worked Example: Find the domain of f(x) = √(x − 6).
The expression under the root must be at least 0: x − 6 ≥ 0, so x ≥ 6.
Domain: [6, ∞)
1. What is the domain of g(x) = (x + 1) / (x − 7)?
Explanation: The denominator is 0 when x = 7, so 7 is excluded. The numerator being 0 at x = −1 is fine. The domain is all real numbers except 7.
2. What is the domain of h(x) = √(2x + 10)?
Explanation: Require 2x + 10 ≥ 0. Then 2x ≥ −10, so x ≥ −5, which is the interval [−5, ∞).
3. For f(x) = (x − 2)(x + 6), what are the zeros and the y-intercept?
Explanation: Set each factor to 0: x = 2 and x = −6. For the y-intercept, f(0) = (0 − 2)(0 + 6) = (−2)(6) = −12, giving (0, −12).
3. Increasing, Decreasing, and Concavity
A function is increasing on an interval if its outputs rise as inputs increase, and decreasing if its outputs fall. Concavity describes how the graph bends. A graph is concave up where its rate of change is increasing, and concave down where its rate of change is decreasing. With a table of equally spaced inputs, compute the change in output between neighbors, then ask whether those changes get larger or smaller.
Worked Example: A table gives f(0) = 10, f(1) = 6, f(2) = 3, f(3) = 1. Describe f.
The outputs fall, so f is decreasing. The changes are −4, −3, −2, which are getting larger (closer to 0).
Decreasing and concave up.
1. A table gives f(1) = 2, f(2) = 5, f(3) = 9, f(4) = 14. Which description fits?
Explanation: The outputs rise, so f is increasing. The changes are 3, 4, 5, which are increasing, so the graph is concave up.
2. A table gives g(0) = 20, g(1) = 18, g(2) = 15, g(3) = 11. Which description fits?
Explanation: The outputs fall, so g is decreasing. The changes are −2, −3, −4, which keep getting smaller, so the rate of change is decreasing and the graph is concave down.
3. A table gives h(0) = 1, h(2) = 9, h(4) = 15, h(6) = 19. Which description fits?
Explanation: The outputs rise, so h is increasing. Over equal steps of 2, the changes are 8, 6, 4, which are shrinking, so the rate of change is decreasing and the graph is concave down.
4. Average Rate of Change
The average rate of change of f from x = a to x = b is the change in output divided by the change in input: f(b) − f(a)b − a. It equals the slope of the secant line through the two points. Always subtract in the same order on top and bottom, and keep the units (such as miles per hour or dollars per week).
Worked Example: Find the average rate of change of f(x) = x² + 2x from x = 1 to x = 4.
f(1) = 1 + 2 = 3 and f(4) = 16 + 8 = 24.
24 − 34 − 1 = 213 = 7
1. A plant is 12 cm tall on day 3 and 33 cm tall on day 10. What is its average growth rate between those days?
Explanation: The change in height is 33 − 12 = 21 cm over 10 − 3 = 7 days, so the rate is 21 ÷ 7 = 3 cm per day.
2. What is the average rate of change of g(x) = x² − 4x from x = 1 to x = 5?
Explanation: g(1) = 1 − 4 = −3 and g(5) = 25 − 20 = 5. The rate is (5 − (−3)) / (5 − 1) = 8 / 4 = 2.
3. An account holds $1,200 in week 2 and $900 in week 8. What is the average rate of change of the balance?
Explanation: The change is 900 − 1200 = −300 dollars over 8 − 2 = 6 weeks, so the rate is −300 ÷ 6 = −50 dollars per week. The negative sign shows the balance is falling.
5. Estimating the Rate of Change at a Point
A secant line needs two points, so an average rate of change cannot be found at a single point. To estimate the rate of change at a point, use the average rate of change over a very small interval containing that point. With a table, use the closest inputs on either side. With an equation, shrink the interval from both sides and watch which value the slopes approach.
Worked Example: A table shows population P at t = 0, 2, 3, 5, 6 hours as 100, 130, 150, 200, 215. Estimate the rate at t = 4.
The nearest inputs around 4 are t = 3 and t = 5.
200 − 1505 − 3 = 502 = 25 per hour
1. Volume V is recorded at t = 1, 4, 5, 9 as 12, 30, 40, 64. Which is the best estimate of the rate at t = 7?
Explanation: The closest inputs around t = 7 are t = 5 and t = 9. The average rate is (64 − 40) / (9 − 5) = 24 / 4 = 6.
2. For f(x) = x², the slope of the secant from x = 2.9 to x = 3 is 5.9, and from x = 3 to x = 3.1 it is 6.1. What value do the slopes approach at x = 3?
Explanation: The left-side slope is below 6 and the right-side slope is above 6, and both move toward 6 as the interval shrinks. A check with x = 2.99 and 3.01 gives (9.0601 − 8.9401) / 0.02 = 6.
3. Secant slopes into x = 2 from the left are 4.9, 4.99, 4.999, and from the right are 5.1, 5.01, 5.001. What is the estimated rate of change at x = 2?
Explanation: Slopes from both sides close in on 5, so the estimated rate of change at x = 2 is 5.
6. Rates of Change in Linear Functions
A linear function has a constant average rate of change over every interval, and that constant is its slope m in y = mx + b. This gives a test for tables with equally spaced inputs: if every consecutive rate matches, the data are linear. A positive slope means the output rises, a negative slope means it falls, and because the rate never changes, a line is neither concave up nor concave down.
Worked Example: Is the table x = 0, 1, 2, 3 with y = 5, 8, 11, 14 linear?
The rates are 3, 3, 3. They match, so the slope is 3, and the y-intercept is 5.
Linear: y = 3x + 5
1. A table gives (1, 4), (3, 10), (5, 16), (7, 22). What does it show?
Explanation: Each step of 2 in x adds 6 to y, so every rate is 6 / 2 = 3. The rate is constant, so the data are linear with slope 3.
2. A table gives (0, 2), (1, 3), (2, 5), (3, 8). What does it show?
Explanation: The rates are 1, 2, 3, which are not constant, so the data are not linear. The rate is increasing, which matches a concave up shape.
3. A linear function passes through (2, 11) and (6, 3). Which equation describes it?
Explanation: The slope is (3 − 11) / (6 − 2) = −8 / 4 = −2. Then y − 11 = −2(x − 2) gives y = −2x + 15. Check with x = 6: −12 + 15 = 3.
7. Rates of Change in Quadratic Functions
A quadratic function f(x) = ax² + bx + c with a ≠ 0 does not have a constant rate of change, because a parabola has no straight pieces. But when inputs are equally spaced, the rates of change themselves change by a constant amount. If that constant is positive, the parabola is concave up (a > 0). If it is negative, it is concave down (a < 0). If the rates change by uneven amounts, the function is neither linear nor quadratic.
Worked Example: For f(x) = 2x², find the rates between x = 0, 1, 2, 3.
The outputs are 0, 2, 8, 18, so the rates are 2, 6, 10.
The rates grow by a constant 4, so f is quadratic and concave up.
1. A table gives g(−1) = 0, g(0) = 1, g(1) = 0, g(2) = −3. What is true of g?
Explanation: The rates are 1, −1, −3. They change by a constant −2, so g is quadratic, and because that change is negative, its parabola is concave down.
2. A table gives h(0) = 1, h(1) = 2, h(2) = 4, h(3) = 8. What is true of h?
Explanation: The rates are 1, 2, 4. They are not constant, so h is not linear. They change by 1, then 2, which is not constant, so h is not quadratic either.
3. A table gives k(0) = 3, k(1) = 1, k(2) = 3, k(3) = 9. What is true of k?
Explanation: The rates are −2, 2, 6. They are not constant, but they grow by a constant 4, so k is quadratic and concave up.
Common Mistakes to Avoid
- Subtracting in opposite orders. Using f(b) − f(a) on top but a − b on the bottom flips the sign of the answer. Use the same order for both.
- Confusing a repeated output with a repeated input. Two inputs may share an output, but one input with two outputs breaks the function rule.
- Calling a decreasing function concave down automatically. Increasing/decreasing is about whether the outputs rise or fall. Concavity is about whether the rates of change rise or fall.
- Using one point to find a rate. A rate of change needs two points, so at a single point you estimate it with a very small interval around it.
- Calling data linear after checking only one pair. Every consecutive rate must match before the data can be called linear.
- Dropping the sign or units. A negative rate means the quantity is falling, and the units (per hour, per week) belong in the answer.
Frequently Asked Questions
What is the difference between average and instantaneous rate of change?
An average rate of change uses two points and is the slope of a secant line. An instantaneous rate of change happens at one moment, and in AP Precalculus you estimate it by taking average rates over tiny intervals around that moment.
How can I tell if a table is linear, quadratic, or neither?
With equally spaced inputs, find the rates between neighbors. If they are all equal, the data are linear. If the rates differ but change by a constant amount, the data are quadratic. If neither pattern holds, it is neither.
Does a positive rate of change mean the graph is concave up?
No. A positive rate means the function is increasing. Concave up means the rate of change itself is increasing. A function can be increasing and concave down at the same time.
Where can I practice more problems like these?
The Rate of Change quiz in the AP Precalculus QBank gives you more practice on this topic, and you can browse every other topic in the full AP Precalculus QBank.