Unit 1 · Topic 1.3 · about 30 minutes · 7 parts · 12 scored checks, 2 of them multiple choice

Rates of Change in Linear and Quadratic Functions

Built from College Board's course framework · Every answer key independently checked

Graph of y = x squared from x = 0 to x = 4 with four secant lines over the intervals 0 to 1, 1 to 2, 2 to 3 and 3 to 4. Their slopes, 1, 3, 5 and 7, are labeled and go up by 2 each step.

Part 1 Think first

Twice as fast, how far to stop?

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

On a dry road, a car going 30 miles per hour needs about 75 feet to stop. About how far does the same car need at 60 miles per hour?

In short. Over equal-length input steps, a linear function's average rates of change stay the same, while a quadratic function's average rates of change change by the same amount each step.

The average rate of change of $f$ from $a$ to $b$ is the slope of the secant line through $(a, f (a))$ and $(b, f (b))$ .
Linear: the average rates over consecutive equal-length intervals are constant, so they change at a rate of zero. Quadratic: they form a linear pattern, so they change at a constant rate.
When those average rates are increasing, the graph is concave up; when they are decreasing, it is concave down.

Part 2 Learn it

Four short ideas, each checked as you go

2a. An average rate of change is a slope

The average rate of change of $f$ over the closed interval $[a, b]$ is

\frac{f (b) - f (a)}{b - a}

which is exactly the slope of the line through the two points $(a, f (a))$ and $(b, f (b))$ on the graph. A line through two points of a graph is called a secant line.

So every average rate of change has a picture. A steep secant line means the output changed a lot for the change in input; a secant line sloping down means the average rate is negative. The diagram at the top of this lesson shows four secant lines on the graph of $y = x^{2}$ .

Quick check · Calculate

No calculator. Let $f (x) = x^{2} - 2 x$ . What is the slope of the secant line through $(1, f (1))$ and $(4, f (4))$ ?

Your answer

2b. Linear functions: one constant rate

For a linear function, the average rate of change over any interval, of any length, is the same number: the slope of the line.

Take a taxi that charges 3 dollars plus 2 dollars per mile, $C (m) = 3 + 2 m$ . Over miles 0 to 1, 1 to 2, 2 to 3 and so on, the average rates are 2, 2, 2: always 2 dollars per mile.

Think of those average rates as a list of their own. For a linear function the list is constant, so the average rates themselves are changing at a rate of zero.

Quick check · Fill the statement

No calculator. Tap a blank, then tap a word.

For a linear function, the average rates of change over consecutive equal-length intervals are , so those average rates are changing at a rate of .

2c. Quadratic functions: rates that change at a constant rate

Now take $q (x) = x^{2}$ at $x = 0, 1, 2, 3, 4$ . The outputs are 0, 1, 4, 9, 16, and the average rates over the consecutive intervals of length 1 are 1, 3, 5, 7.

Those rates are not constant, but they go up by the same amount, 2, every step. They follow a linear pattern. That is true of every quadratic function: over consecutive equal-length intervals, its average rates of change can be given by a linear function, so they are changing at a constant rate.

The same test works on a sequence, a list of outputs at equally spaced inputs. Find the differences between consecutive terms. If they are constant, the pattern is linear. If the differences of those differences are constant and not zero, it is quadratic.

Quick check · Calculate

No calculator. A quadratic function $p$ has average rates of change 5, 2 and $- 1$ over the intervals $[0, 1]$ , $[1, 2]$ and $[2, 3]$ . What is its average rate of change over $[3, 4]$ ?

Your answer

2d. Concavity from the rates

In 1.1, concave up meant the rate of change was increasing. Now you can test it with numbers.

If the average rates over equal-length intervals keep increasing for all small intervals, the graph is concave up.
If they keep decreasing, the graph is concave down.

For $q (x) = x^{2}$ the rates 1, 3, 5, 7 increase, so the graph is concave up. Watch the signs: the rates $- 9$ , $- 5$ , $- 2$ are negative but increasing, so that graph is falling and concave up.

Quick check · Sort it

No calculator. Each list gives a function's average rates of change over consecutive intervals of equal length, left to right. Tap a list, then tap the shape of the graph.

Concave up

Concave down

Part 3 Watch it done

Stopping distance, ten miles per hour at a time

Watch each step, then do the last one yourself.

A driving safety guide gives a car's stopping distance $d (s)$ , in feet, at speed $s$ miles per hour:

$d (20) = 40$ , $d (30) = 75$ , $d (40) = 120$ , $d (50) = 175$ , $d (60) = 240$ .

Decide what kind of function fits the table, and use it to estimate $d (70)$ .

Step 1 · Average rates
Over each 10 mile per hour step: $\frac{75 - 40}{10} = 3.5$ , then 4.5, 5.5 and 6.5 feet per mile per hour.
Step 2 · How the rates change
The rates go up by exactly 1 each step. Average rates over equal intervals that change at a constant rate are the mark of a quadratic function. They are increasing, so the graph is concave up.
Why this step?
Why can a linear model not fit this table?
A linear function has the same average rate of change over every interval. Here the rates are 3.5, 4.5, 5.5 and 6.5, so no single slope gives all of them.
Your step · Calculate

No calculator. Finish it. Estimate the stopping distance at 70 miles per hour, in feet.

Your answer feet

Part 4 Try it

Linear, quadratic or neither?

Try it · Match

No calculator. Each list gives the outputs of a function at $x = 0, 1, 2, 3$ . Tap a list, then the kind of function it could come from.

Practice

Practice until it is automatic

More average rates of change, which are secant slopes, for quadratic and cubic functions.

Average rate of change practice page

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page.

Free response is 37.5% of the exam score. This one is shaped like Question 1, Function Concepts.

Free-response practice · Question 1 style

Graphing calculator required. A function $g$ has $g (0) = 3$ , $g (2) = 7$ , $g (4) = 15$ , $g (6) = 27$ and $g (8) = 43$ . Explain whether these values are consistent with a linear function or a quadratic function, and say whether the graph of $g$ is concave up or concave down, using average rates of change.

Your answer stays on this page. It is never sent or saved.

Model answer

The average rates of change over $[0, 2]$ , $[2, 4]$ , $[4, 6]$ and $[6, 8]$ are 2, 4, 6 and 8. They are not constant, so the values are not consistent with a linear function. They increase by 2 over each consecutive interval of length 2, a constant rate, which is consistent with a quadratic function. Because the average rates are increasing, the graph of $g$ is concave up.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Computes the average rates of change over the equal intervals: 2, 4, 6 and 8.
Rules out linear because the rates are not constant.
Identifies quadratic because the rates change by the same amount each interval.
Says concave up because the average rates are increasing.

Part 6 Check yourself

A growing tile pattern

The two case questions use the same data. Answer them from the table.

An artist builds a pattern of figures from square tiles. $T (n)$ is the number of tiles in figure $n$ .

Tiles in each figure
Figure $n$	Tiles $T (n)$
1	2
2	6
3	12
4	20
5	30

Case question 1 · Calculate

No calculator. What is the average rate of change of $T$ from $n = 2$ to $n = 5$ , in tiles per figure?

Your answer

Case question 2 · Multiple choice

No calculator. Which statement describes the average rates of change of $T$ over consecutive intervals of length 1?

The rest stand on their own and do not use the case.

Check yourself · Spot the error

No calculator. A student tests whether $h$ is linear, where $h (0) = 1$ , $h (1) = 3$ , $h (2) = 7$ and $h (3) = 13$ . Tap the line with the mistake.

Check yourself · Decide and justify

No calculator. A seedling's average rates of growth over weeks 0 to 2, 2 to 4 and 4 to 6 are 3.0, 2.2 and 1.4 centimeters per week. What shape is the graph of its height?

Which reason supports your answer?

Part 7 Keep it

Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 1.2 Rates of Change · Calculate

No calculator. For the falling stone, $d (1.9) = 17.689$ , $d (2) = 19.6$ and $d (2.1) = 21.609$ , in meters. Use the interval from $t = 1.9$ to $t = 2.1$ to estimate the rate of change of $d$ at $t = 2$ , in meters per second.

Your answer m/s

From 1.1 Change in Tandem · Multiple choice

No calculator. A function $f$ has $f (1) = 4$ , $f (2) = 7$ , $f (3) = 9$ , $f (4) = 6$ and $f (5) = 2$ . Between these inputs its graph is made of straight segments. On which interval is $f$ decreasing?

In three lines

The average rate of change over $[a, b]$ is the slope of the secant line through $(a, f (a))$ and $(b, f (b))$ .
Over consecutive equal intervals, linear rates are constant; quadratic rates change by the same amount each step.
Increasing average rates mean concave up; decreasing average rates mean concave down.

Vocabulary

Secant line: A line through two points on the graph of a function. Its slope is the average rate of change between them.
Linear function: A function whose average rate of change is the same over every interval.
Quadratic function: A function whose average rates of change over consecutive equal-length intervals change at a constant rate.
Sequence: A list of output values, often at equally spaced inputs, whose differences can be studied like average rates of change.
Concave up: The average rates of change over equal-length intervals are increasing.
Concave down: The average rates of change over equal-length intervals are decreasing.

Course alignment, for teachers

AP Precalculus topic 1.3, Unit 1: Polynomial and Rational Functions.

Skill 3.B: Apply numerical results in a given mathematical or applied context.
Skill 3.C: Support conclusions or choices with a logical rationale or appropriate data.

This lesson maps all 6 essential knowledge statements of the topic to the part that teaches each one: 6 are checked by a scored question. The map is in the lesson file.