Knowledge Notes | Carey AP Precalculus Practice Lab

For equal input steps, first differences support a linear model and constant nonzero second differences support a quadratic model.

What a strong response shows

Check equal input spacing before subtracting.
State whether first or second differences are constant.
Use a nonzero second difference for a quadratic model.

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What You Need to Know

Key idea

Linear functions change by a constant amount for equal input steps; quadratic functions change by a changing amount.

Fast decision rule

For equally spaced inputs, check first differences before choosing a model.

Worked example

A table that adds $4$ each time has a constant rate. A table whose first differences change by $2$ has a constant second difference and suggests a quadratic model.

Common mistake

Calling a table linear because the outputs increase.

Mini check

What must remain constant for equally spaced linear data?

Learning Objectives

Key idea

1.3.A finds average rates for sequences and functions; 1.3.B describes how those rates change.

Fast decision rule

Compute the rate on each requested interval before describing the pattern.

Worked steps

$1$ . Keep input spacing visible. $2$ . Find first differences. $3$ . Divide by the input step if needed. $4$ . Compare the rates or their changes.

Common mistake

Comparing first differences when the input steps are not equal.

Mini check

Why can a first-difference pattern be misleading when the x-steps differ?

Key Concepts

Key idea

A constant first difference signals a constant rate; a constant second difference signals a quadratic pattern for equal steps.

Fast decision rule

First differences answer “how much output changes”; second differences answer “how that change changes.”

Worked example

If first differences are $3$ , $5$ , $7$ , the second differences are $2$ and $2$ . The rate is increasing by a constant amount.

Common mistake

Calling the second difference the average rate itself.

Mini check

If the second difference is zero, what model behavior should you expect?

Relevant Representations

Key idea

A sequence, table, graph, and formula reveal the same linear or quadratic pattern from different angles.

Fast decision rule

Use the table to calculate differences, the graph to see straightness or curvature, and the formula to verify the model.

Match the evidence to the representation
Representation	Evidence	Interpretation
Sequence	equal input steps	compare differences
Table	first and second differences	identify model type
Graph	line or parabola shape	check curvature
Formula	degree and coefficients	predict rate behavior

Common mistake

Using a graph’s visual smoothness instead of checking the data pattern.

Mini check

Which representation makes second differences fastest to compute?

Essential Relationships or Formulas

Key idea

Differences and average rates connect a table to the behavior of its model.

\Delta f=f(x+h)-f(x)

\operatorname{AROC}_{[x,x+h]}=\frac{f(x+h)-f(x)}{h}

\Delta^2 f=\Delta f_{\text{next}}-\Delta f_{\text{current}}

Fast decision rule

Use differences only after confirming the input spacing.

Worked example

For equally spaced inputs, first differences $6$ , $10$ , $14$ give second difference $4$ . The average rate increases from one interval to the next.

Common mistake

Forgetting to divide a difference by $h$ when the input step is not $1$ .

Mini check

If $h$ doubles but the output change stays fixed, what happens to the average rate?

Worked Examples

Key idea

Make a small difference table before deciding whether the function is linear or quadratic.

Fast decision rule

Write the x-step above the table, then label each difference with its order.

Worked steps

$1$ . List consecutive inputs. $2$ . Subtract consecutive outputs. $3$ . Repeat for second differences. $4$ . Describe the pattern. $5$ . Check one formula value.

Worked example

Outputs $2$ , $6$ , $12$ , $20$ at equal input steps have first differences $4$ , $6$ , $8$ and second difference $2$ , so the pattern is quadratic-like.

Common mistake

Stopping after one difference when the question asks how the rate changes.

Mini check

What does a positive constant second difference say about the rate?

Common Mistakes

Key idea

The usual traps are uneven steps, reversed subtraction, and confusing output change with rate.

Fast decision rule

Keep a separate row for input step, first difference, and second difference.

Worked example

If the x-steps are $1$ and $2$ , a vertical change of $8$ cannot be compared directly with a vertical change of $8$ over the other step.

Common mistake

Treating equal output changes as equal rates when the input intervals differ.

Mini check

For equally spaced inputs, which row of your difference table supports a quadratic model, and why must its constant second difference be nonzero?

AP Exam Tips

Key idea

AP tables often give just enough values to reveal a rate pattern.

Fast decision rule

Check spacing, compute differences, and state whether the rate is constant, increasing, or decreasing.

Worked steps

$1$ . Confirm equal x-steps. $2$ . Compute first differences. $3$ . If needed, compute second differences. $4$ . Translate the pattern into words or a model. $5$ . Verify a given value.

Common mistake

Choosing a quadratic only because the graph is curved.

Mini check

Can you explain your model choice using a difference row?

Practice

Use the Questionbank items below to practice linear and quadratic rate patterns.

TRY IT IN A NEW SETTING

Can you transfer the method?

At inputs $0$ , $2$ , $4$ , $6$ , outputs are $1$ , $5$ , $17$ , $37$ . Give the difference evidence and one matching quadratic.

Compare your reasoning

First differences are $4$ , $12$ , $20$ ; second differences are the nonzero constant $8$ for equal input steps of $2$ . The quadratic $y=x^{2}+1$ fits all four values.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.3 · Rates of Change in Linear and Quadratic Functions

Topic 1.3 · Rates of Change in Linear and Quadratic Functions

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?