Unit 1 · Topic 1.3

1.3 Rates of Change in Linear and Quadratic Functions

Linear and quadratic functions have very predictable rates of change. A linear function's rate never changes, while a quadratic's rates over equal steps change by the same amount every time. Spotting these patterns in a table tells you which kind of function you're looking at and how its graph bends.

Key terms

secant line
slope
constant rate of change
second differences
concavity

Secant lines and average rate of change

A secant line connects two points on a graph. The average rate of change of f over [a, b] is exactly the slope of the secant line from (a, f(a)) to (b, f(b)).

This gives you a picture: a steep secant line means a large average rate, and a secant line that falls to the right means a negative one. It also works for sequences, where you find the change between terms divided by the change in position.

Linear functions: constant rate

For a linear function f(x) = mx + b, the average rate of change over any interval, of any length, is the slope m. That is why its graph is a straight line.

Over consecutive equal-length intervals, the average rates are all the same number. Their rate of change is zero: the rates never change. In a table with equally spaced inputs, the first differences (the changes in output) are constant.

Quadratic functions: rates that change steadily

For a quadratic f(x) = ax² + bx + c, the average rates of change over consecutive equal-length intervals form a linear pattern: they go up or down by the same amount each time.

In a table with equally spaced inputs, the first differences change by a constant amount, so the second differences (the differences of the differences) are constant. If the inputs go up by 1, the second difference equals 2a.

Because the rates change at a constant rate, a quadratic's graph bends the same way everywhere. If the rates keep increasing (a > 0), the graph is concave up; if they keep decreasing (a < 0), it is concave down.

Concavity from average rates

This idea works for any function, not just quadratics. If the average rate of change over small equal-length intervals keeps increasing, the graph is concave up there. If it keeps decreasing, the graph is concave down.

So a table can tell you concavity: compute the average rates over consecutive equal intervals and see whether they grow or shrink.

Function type	First differences (equal steps)	Second differences	Graph
Linear	Constant	Zero	Straight line
Quadratic	Change by a constant amount	Constant, not zero	Parabola, one concavity

Worked examples

Try each one yourself first, then open the solution.

Example 1
Identifying a quadratic from a table
A function f has f(0) = 3, f(1) = 5, f(2) = 11, f(3) = 21 and f(4) = 35. Explain why a quadratic fits these values, describe the concavity, and find the quadratic.
Show the solutionHide the solution
1. Step 1: The inputs are equally spaced (steps of 1). First differences: 5 − 3 = 2, 11 − 5 = 6, 21 − 11 = 10, 35 − 21 = 14.
2. Step 2: Second differences: 6 − 2 = 4, 10 − 6 = 4, 14 − 10 = 4. They are constant and not zero, so the average rates change at a constant rate: that is the quadratic pattern.
3. Step 3: The average rates over consecutive intervals (2, 6, 10, 14) are increasing, so the graph is concave up.
4. Step 4: With steps of 1, the second difference is 2a, so 2a = 4 and a = 2. Since f(0) = 3, c = 3. Then f(1) = 2 + b + 3 = 5 gives b = 0.
5. Step 5: Check: f(3) = 2(9) + 3 = 21 and f(4) = 2(16) + 3 = 35. Both match.
Answer: f(x) = 2x² + 3. It is concave up because its average rates of change over equal intervals keep increasing.
Example 2
Equation of a secant line
Let f(x) = x² − 4x + 1. Find the equation of the secant line through the points where x = 1 and x = 4.
Show the solutionHide the solution
1. Step 1: Find the points: f(1) = 1 − 4 + 1 = −2 and f(4) = 16 − 16 + 1 = 1. The points are (1, −2) and (4, 1).
2. Step 2: Slope = average rate of change = (1 − (−2)) / (4 − 1) = 3/3 = 1.
3. Step 3: Point-slope form with (1, −2): y = −2 + 1(x − 1), which simplifies to y = x − 3.
Answer: y = x − 3 (equivalently y = −2 + 1(x − 1)).
Example 3
Trap: unequal input steps
A table shows g(0) = 4, g(1) = 7, g(3) = 13 and g(6) = 22. A student says the output changes are 3, 6 and 9, so g is not linear. Is the student right?
Show the solutionHide the solution
1. Step 1: The inputs are not equally spaced: the steps are 1, 2 and 3. Comparing raw output changes only works when the input steps are equal.
2. Step 2: Compute the average rates instead: 3/1 = 3, 6/2 = 3 and 9/3 = 3.
3. Step 3: The average rate of change is 3 on every interval, which is exactly the linear pattern. With g(0) = 4, the function is g(x) = 3x + 4.
Answer: No. The rates are all 3, so the data fit the linear function g(x) = 3x + 4.

Common mistakes

Using first or second differences when the inputs are not equally spaced. Divide each change in output by the change in input first.
Calling data quadratic because the first differences increase. They must increase by the same amount each time (constant second differences).
Mixing up which differences tell you concavity. Increasing average rates mean concave up, even if the outputs themselves are decreasing.

On the exam

Expect tables where you must decide whether data are linear, quadratic or neither, with a reason that mentions how the average rates of change behave over equal intervals.
A good justification sounds like: “Over consecutive equal-length intervals, the average rates of change increase by the same amount, so a quadratic model is appropriate.”

Connected topics

Videos

AP Precalculus – 1.3 Rates of Change Linear and Quadratic Functions
The AlgebrosWatch on YouTube (opens in a new tab)
Rates of Change in Quadratic Functions [AP Precalculus Topic 1.3]
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)
Finding Rates of Change in Under 3 mins (AP Precalculus Topic 1.3)
Maximum InsightWatch on YouTube (opens in a new tab)
AP Precalculus Notes (Topic 1.3) Rates of Change in Linear and Quadratic Functions
Mr. SindelWatch on YouTube (opens in a new tab)
1.3A - Rates of Change in Linear and Quadratic Functions [AP Precalculus]
MrHelpfulNotHurtfulWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 1.3 Rates of Change in Linear and Quadratic Functions. Pick an answer to see if you got it, and why.

x	0	2	4	6
f(x)	5	9	21	41

Table of values

Question 1 of 4

What is the value of f(8)?

Question 2 of 4

Let f(x) = ax² + bx + c. Over three consecutive intervals of length 1, the average rates of change of f are 5, 2 and −1, in that order. What is the value of a?

x	0	1	2	3	4
f(x)	3	4	7	12	19

Table of values

Question 3 of 4

What is the average rate of change of f over the interval 1 ≤ x ≤ 4?

Question 4 of 4

Which statement about f is best supported by the table?

0 of 4 answered