Topic 1.3

Rates of Change in Linear and Quadratic Functions

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01
Linear function: rate of change = slope = $m$ (always the same number)
02
Quadratic function: 2nd differences are constant
03
Linear: the average rate of change over consecutive equal-length input-value intervals is constant.

Quadratic: the average rate of change over consecutive equal-length input-value intervals is linear (it changes by the same amount each time).

Linear function: rate of change = slope = $m$ (always the same number)
Quadratic function: 2nd differences are constant
Linear: the average rate of change over consecutive equal-length input-value intervals is constant.

Quadratic: the average rate of change over consecutive equal-length input-value intervals is linear (it changes by the same amount each time).
average rate of change $=$ 1st difference $\div$ spacing
$\frac{\Delta y}{\Delta x} = \frac{\text{1st difference}}{h}$
Average rates of change over consecutive equal-length intervals are increasing → the graph is concave up.

Average rates of change over consecutive equal-length intervals are decreasing → the graph is concave down.
For a quadratic: sign of the 2nd difference $=$ sign of $a$ .

Positive → concave up. Negative → concave down.
To test whether a function is linear, compute $\dfrac{\Delta y}{\Delta x}$ on each interval. Equal spacing is not required.

If all of those slopes match, the function is linear. If they do not, it is not linear.
A degree-n polynomial has constant $n$ -th differences.

Practice 23 problems Next: Chapter 1.4 Extrema and Rates of Change

Linear Functions: Rate of Change = Slope

Remember from 1.2: the average rate of change is the slope of the secant line.

For a linear function $f(x)$ = mx + b, every secant line has the exact same slope — the number $m$ . So the rate of change between ANY two points is always $m$ .

KEY RULE

Linear function: rate of change = slope = $m$ (always the same number)

CONCEPT

Why This Makes Sense

Rate of change is just another name for slope (from 1.2).

A line has ONE slope — it doesn't bend or curve — so no matter which two points you pick, you get the same slope, and therefore the same rate of change.

Quadratic Functions: the Rate of Change ITSELF Changes

For a quadratic function, the rate of change is different on every interval — it's not constant like a line. But here's the key discovery: if you look at HOW the rate of change itself is changing, THAT turns out to be constant.

Example 1: f(x) = x²

The 1st differences (1, 3, 5, 7) are not constant — that's the changing rate of change. But the 2nd differences are constant at 2.

Example 2: f(x) = 2x² + 3x + 1

Same pattern, different quadratic: the 2nd differences are constant at 4. This isn't a coincidence — it happens for every quadratic function.

KEY RULE

Quadratic function: 2nd differences are CONSTANT

Quick check

A table with $x$ spaced by $1$ has outputs $3, 5, 9, 15, 23$ . Linear, quadratic, or neither?

A Bigger Rule: x Doesn't Have to Increase by 1

Every example so far used x-values spaced 1 apart (0, 1, 2, 3, 4). But the difference method works just as well as long as $x$ increases by the SAME amount each time — even if that amount isn't 1.

Worked example

$f(x)$ has these values. $x$ increases by 2 each time. Is this quadratic?

Even with $x$ spaced by 2 instead of 1, the 2nd differences are still perfectly constant at 4 → this function is quadratic.

CONCEPT

The Real Rule

$x$ must increase by a CONSTANT amount each step (called equal spacing) — that amount can be 1, 2, 0.5, or anything else, as long as it never changes.

If x-values are NOT equally spaced, the difference method does not work directly.

The Words the Exam Actually Uses

The differences method is how you WORK the problem. But the answer choices on the exam are written a different way, and you have to recognize your own answer when you see it.

KEY RULE

Linear: the average rate of change over consecutive equal-length input-value intervals is CONSTANT.

Quadratic: the average rate of change over consecutive equal-length input-value intervals is LINEAR (it changes by the same amount each time).

Read those two lines again. "Constant rate of change" is the exam's way of saying "1st differences are constant". "The rate of change is itself linear" is the exam's way of saying "2nd differences are constant".

CONCEPT

Why the Two Descriptions Match

For a linear function the slope never changes, so every interval of the same width gives the same average rate of change.

For a quadratic the slope changes — but it changes at a steady pace, so the list of average rates of change is itself an arithmetic (linear) list.

The Piece That Trips People Up

A 1st difference and an average rate of change are only the SAME number when $x$ moves by exactly $1$ .

KEY RULE

average rate of change $=$ 1st difference $\div$ spacing

\frac{\Delta y}{\Delta x} = \frac{\text{1st difference}}{h}

Go back to the spacing- $2$ example in section 3. Its 1st differences were $-3$ , $1$ , $5$ , $9$ — but those are NOT the rates of change. Divide each by the spacing $h = 2$ :

-1.5, \quad 0.5, \quad 2.5, \quad 4.5

Those four numbers are the average rates of change. Notice they still go up by the same amount ( $2$ ) each time — still linear, so still a quadratic. The differences method gave the right ANSWER, but only the divided numbers are the right rates.

COMMON MISTAKE

Reporting a 1st difference as "the rate of change" when the spacing is not $1$ . On a spacing- $2$ table that answer is exactly twice too big.

Quick check

On a table where $x$ goes up by $2$ each row, the first differences are all $6$ . What is the rate of change?

Rates of Change and Concavity

The same list of average rates tells you which way the graph bends.

KEY RULE

Average rates of change over consecutive equal-length intervals are INCREASING → the graph is concave up.

Average rates of change over consecutive equal-length intervals are DECREASING → the graph is concave down.

For a quadratic $f(x) = ax^2 + bx + c$ there is an even faster route: with spacing $h$ , every 2nd difference equals $2ah^2$ . Since $h^2$ is positive, the 2nd difference has the SAME SIGN as $a$ .

KEY RULE

For a quadratic: sign of the 2nd difference $=$ sign of $a$ .

Positive → concave up. Negative → concave down.

Worked example

A table gives $f$ at $x = 0, 2, 4, 6$ with values $30, 22, 16, 12$ . Is $f$ concave up or concave down?

01
1st differences:
$22 - 30 = -8 \qquad 16 - 22 = -6 \qquad 12 - 16 = -4$
02
Divide by the spacing $h = 2$ to get the average rates of change:
$-4, \quad -3, \quad -2$
03
The rates are $-4 \to -3 \to -2$ — increasing (they are climbing toward zero). So the graph is concave up.
04
Check with 2nd differences: $-6 - (-8) = 2$ and $-4 - (-6) = 2$ . Constant and positive → quadratic, and $a > 0$ → concave up. Same answer.

COMMON MISTAKE

Reading "the values are going down, so it must be concave down." Direction and concavity are independent — here $f$ decreases the whole way and is still concave up.

When the Spacing Is NOT Equal

Section 3 said the differences method needs equal spacing. That is true — but it does not mean you are stuck.

KEY RULE

To test whether a function is LINEAR, compute $\dfrac{\Delta y}{\Delta x}$ on each interval. Equal spacing is not required.

If all of those slopes match, the function is linear. If they do not, it is not linear.

Worked example

Is $f$ linear? $f(1) = 5$ , $f(4) = 14$ , $f(9) = 29$ .

01
The spacing is $3$ then $5$ — not equal, so do not take raw differences.
02
Compute a slope on each interval:
$\frac{14 - 5}{4 - 1} = \frac{9}{3} = 3 \qquad \frac{29 - 14}{9 - 4} = \frac{15}{5} = 3$
03
Both slopes are $3$ , so $f$ is linear — even though the table is unevenly spaced.

COMMON MISTAKE

Choosing "cannot be determined" because the spacing is uneven. For the LINEAR question you can always decide. It is the higher-degree questions (2nd, 3rd differences) that genuinely need equal spacing.

Extending the Pattern: Cubic, Quartic, and Beyond

The same idea keeps going. If a quadratic's 2nd differences are constant, what about a cubic or quartic?

Cubic Example: f(x) = x³

For this cubic, the 1st and 2nd differences both change — but the 3rd differences are constant at 6.

Quartic Example: f(x) = x⁴

Here, only the 4th differences become constant (at 24).

KEY RULE

A degree-n polynomial has CONSTANT $n$ -th differences.

CONCEPT

The Full Pattern

Degree 1 (linear) → 1st differences constant (this is just the slope!)

Degree 2 (quadratic) → 2nd differences constant

Degree 3 (cubic) → 3rd differences constant

Degree 4 (quartic) → 4th differences constant

Degree $n$ → $n$ -th differences constant

CONCEPT

Bonus Fact (for the curious)

When $x$ increases by 1 each time, the constant $n$ -th difference equals $n!$ ( $n$ factorial) times the leading coefficient.

Check it: $x^{4}$ has leading coefficient 1, and $4! = 24$ — matching the constant 4th difference of 24 above!

Step-by-Step: Finding the Degree from a Table

Worked Example A — a Quadratic

Worked example

What is the degree of the function shown in this table?

x	f(x)
0	3
1	7
2	13
3	21
4	31

01
Compute the 1st differences: 7−3=4, 13−7=6, 21−13=8, 31−21=10.
02
1st differences: 4, 6, 8, 10 — NOT constant, so this is not degree 1 (not linear).
03
Compute the 2nd differences: 6−4=2, 8−6=2, 10−8=2.
04
2nd differences: 2, 2, 2 — constant! Since the differences become constant at the 2nd level, this function has degree 2 (it's quadratic).

Worked Example B — a Quartic

Worked example

What is the degree of the function shown in this table?

x	f(x)
0	0
1	3
2	20
3	87
4	264
5	635

01
1st differences: 3−0=3, 20−3=17, 87−20=67, 264−87=177, 635−264=371.
02
1st differences (3, 17, 67, 177, 371) are not constant — keep going.
03
2nd differences: 17−3=14, 67−17=50, 177−67=110, 371−177=194 — still not constant.
04
3rd differences: 50−14=36, 110−50=60, 194−110=84 — still not constant.
05
4th differences: 60−36=24, 84−60=24 — CONSTANT at last!
06
Since the differences first become constant at the 4th level, this function has degree 4 (it's a quartic).

COMMON MISTAKE

Keep taking differences until you actually reach a constant row — stopping at the 1st or 2nd difference too early (assuming it's quadratic) is a common error.

A single non-constant value is enough to rule out that level — you don't need every value to differ, just one.

Degree $n$ needs $n$ differences to become constant — a cubic will NOT show a constant 2nd difference.

The difference method only works when x-values are equally spaced — check the spacing first.

Practice Problems

Worked example

Problem 1. Find the degree: $f$ gives 2, 5, 10, 17, 26 at x = 0,1,2,3,4.

01
1st differences: 3, 5, 7, 9 — not constant.
02
2nd differences: 2, 2, 2 — constant. Degree = 2 (quadratic).

Worked example

Problem 2. Find the degree: $f$ gives 0, 1, 8, 27, 64 at x = 0,1,2,3,4.

01
1st differences: 1, 7, 19, 37 — not constant.
02
2nd differences: 6, 12, 18 — not constant.
03
3rd differences: 6, 6 — constant. Degree = 3 (cubic).

Common slips

Reporting a 1st difference as "the rate of change" when the spacing is not $1$ . On a spacing- $2$ table that answer is exactly twice too big.
Reading "the values are going down, so it must be concave down." Direction and concavity are independent — here $f$ decreases the whole way and is still concave up.
Choosing "cannot be determined" because the spacing is uneven. For the linear question you can always decide. It is the higher-degree questions (2nd, 3rd differences) that genuinely need equal spacing.
Keep taking differences until you actually reach a constant row — stopping at the 1st or 2nd difference too early (assuming it's quadratic) is a common error.

A single non-constant value is enough to rule out that level — you don't need every value to differ, just one.

Degree $n$ needs $n$ differences to become constant — a cubic will not show a constant 2nd difference.

The difference method only works when x-values are equally spaced — check the spacing first.

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3 cards · Rates of change

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Recap card

5 lines to re-read the night before.

01
Linear (degree 1): rate of change = slope, always constant (1st differences constant).
02
Quadratic (degree 2): the rate of change itself changes at a constant pace (2nd differences constant).
03
The pattern extends: degree $n$ ⟺ $n$ -th differences are constant.
04
x-values must be equally spaced (constant step size) for the difference method to work — the step doesn't have to be 1.
05
To find a function's degree from a table, keep taking differences until you hit a constant row — the number of steps is the degree.

Practice 23 problems Next: Chapter 1.4 Extrema and Rates of Change

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Rates of Change in Linear and Quadratic Functions

Linear Functions: Rate of Change = Slope

Quadratic Functions: the Rate of Change ITSELF Changes

Example 1: f(x) = x²

Example 2: f(x) = 2x² + 3x + 1

A Bigger Rule: x Doesn't Have to Increase by 1

The Words the Exam Actually Uses

The Piece That Trips People Up

Rates of Change and Concavity

When the Spacing Is NOT Equal

Extending the Pattern: Cubic, Quartic, and Beyond

Cubic Example: f(x) = x³

Quartic Example: f(x) = x⁴

Step-by-Step: Finding the Degree from a Table

Worked Example A — a Quadratic

Worked Example B — a Quartic

Practice Problems

Recap card

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