Unit 1 · Topic 1.3

Rates of Change in Linear and Quadratic Functions

Use equal-length intervals to recognize constant linear rates, linearly changing quadratic rates, constant second differences, and the connection between changing rates and concavity.

Learning Goals

Determine average rates of change for linear and quadratic functions and sequences.
Explain why every average rate of change of a linear function equals its slope.
Show that quadratic average rates change linearly over consecutive equal-length intervals.
Use first and second differences to distinguish linear and quadratic patterns.
Connect increasing or decreasing small-interval rates to concavity.
Apply numerical results in context and support conclusions with calculations.

1. Why Equal-Length Intervals Matter

A fair comparison keeps the input interval length fixed. For consecutive intervals of length \(h>0\), compare

\[[x,x+h],\quad[x+h,x+2h],\quad[x+2h,x+3h].\]

If interval lengths differ, output changes and average rates may not reveal the underlying pattern clearly. State the interval before reporting each rate.

2. Linear Functions Have One Constant Rate

Let \(L(x)=mx+b\). On any interval \([a,c]\),

\[\frac{L(c)-L(a)}{c-a}=\frac{(mc+b)-(ma+b)}{c-a}=m.\]

The average rate is always \(m\), regardless of the interval's location or length. Therefore the rate of change of the average rates is 0.

3. Linear Pattern in a Table and Sequence

For \(L(x)=2x+1\), equally spaced inputs produce constant first differences.

Input \(x\)	0	1	2	3	4
Output \(L(x)\)	1	3	5	7	9
First differences	\(+2,\ +2,\ +2,\ +2\)
Changes in first differences	\(0,\ 0,\ 0\)

With unit input steps, each first difference equals the average rate. The outputs form an arithmetic sequence with common difference 2.

4. Quadratic Rates Form a Linear Pattern

Let \(Q(x)=Ax^2+Bx+C\). Over an interval of fixed length \(h\), simplify the difference quotient:

\[\begin{aligned}\operatorname{AROC}_{[x,x+h]}&=\frac{Q(x+h)-Q(x)}{h}\\&=2Ax+Ah+B.\end{aligned}\]

For a fixed \(h\), this expression is linear in the starting input \(x\). That is why quadratic average rates are not constant but change at a constant rate.

5. Secant Slopes on a Quadratic

A line keeps one secant slope everywhere, while a quadratic's secant slopes change linearly; increasing small-interval slopes indicate concave-up behavior.

Each equal-length interval creates a secant line. Moving the interval across a parabola changes the secant slope. For a concave-up quadratic, these slopes increase from more negative to less negative to positive.

The graph can be decreasing while concave up: left of the minimum, outputs fall but the negative slopes become less negative.

6. First and Second Differences

Consider \(Q(x)=x^2-4x+1\) at consecutive integer inputs.

Input \(x\)	0	1	2	3	4	5
Output \(Q(x)\)	1	\(-2\)	\(-3\)	\(-2\)	1	6
First differences / unit-interval AROCs	\(-3,\ -1,\ 1,\ 3,\ 5\)
Second differences	\(2,\ 2,\ 2,\ 2\)

The first differences change, so the function is not linear. Their constant difference of 2 identifies the quadratic pattern and shows that the average rates increase consistently.

7. The Interval Length Changes the Numbers, Not the Pattern

For the same quadratic, use consecutive intervals of length 2:

\[\operatorname{AROC}_{[0,2]}=-2,\quad \operatorname{AROC}_{[2,4]}=2,\quad \operatorname{AROC}_{[4,6]}=6.\]

These rates increase by 4 each time. Unit intervals produced a change of 2; length-2 intervals produce a change of 4. In both cases, the average rates follow a linear pattern and their changes are constant.

8. Leading Coefficient and Second Differences

For \(Q(x)=Ax^2+Bx+C\) with unit-spaced inputs, the constant second difference is

\[\Delta^2 Q=2A.\]

If \(A>0\), average rates increase and the graph is concave up.
If \(A<0\), average rates decrease and the graph is concave down.
The coefficients \(B\) and \(C\) affect position and individual rates, but not the unit-step second difference.

9. Concavity Describes Changing Rates

Concavity and increasing/decreasing behavior answer different questions.

Small-interval average rates	Graph behavior
Rates increase as the interval moves right	Concave up
Rates decrease as the interval moves right	Concave down
Rates stay constant	Linear; no upward or downward bending

A concave-up function may be decreasing, and a concave-down function may be increasing. Check the sign of the rate for direction and the change in rates for concavity.

10. Context Example: Vertical Motion

A height model is \(H(t)=-5t^2+20t+2\), where height is measured in meters and time in seconds.

Interval (s)	Average rate (m/s)	Meaning
\([0,1]\)	15	Height increases rapidly.
\([1,2]\)	5	Height still increases, but more slowly.
\([2,3]\)	\(-5\)	Height decreases.
\([3,4]\)	\(-15\)	Height decreases more rapidly.

The rates decrease by 10 m/s across consecutive one-second intervals. This constant rate-of-rate pattern is consistent with a concave-down quadratic.

11. AP Reasoning and Common Errors

A strong conclusion includes the interval length, calculated rates, their pattern, and what the pattern proves.

Response pattern: “Over consecutive intervals of length ___, the average rates are ___. Because these rates increase/decrease by a constant ___, the data are consistent with a quadratic function whose graph is concave up/down.”

Do not compare raw output differences when the input steps are unequal.
Do not call a quadratic's rate constant; its change in rate is constant.
Do not confuse a negative rate with concave down.
Do not infer a quadratic pattern from only one first difference or one average rate.
Include contextual units for both the average rate and its change when the problem supplies units.

Key Takeaways

Compare constant linear rates with the linearly changing average rates and constant rate-of-rate patterns of quadratic functions over equal-length intervals.
Core relationship: \(Q(x)=Ax^2+Bx+C\Rightarrow\operatorname{AROC}_{[x,x+h]}=2Ax+Ah+B\)
Error check: Equal spacing is essential: compare rates over consecutive intervals of the same length before identifying a first- or second-difference pattern.

Checkpoint · Topic 1.3

Compare \(L(x)=3x-2\) and \(Q(x)=2x^2-4x+1\) at integer inputs from 0 through 4.

Create an output table for both functions.
Find each function's average rates over the four consecutive unit intervals.
Find the changes in those average rates and classify each pattern as linear or quadratic.
Determine the concavity of \(Q\) and identify the first listed interval where its average rate exceeds the rate of \(L\).
Write a complete AP-style justification using the numerical evidence.