Knowledge Notes | Carey AP Precalculus Practice Lab

Illustrations compare additive and multiplicative change; they are not numerical plots.

What a strong response shows

Compare equal input increments.
Linear models have equal output differences.
Exponential models have constant positive output ratios.

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What You Need to Know

Key idea

Linear functions add the same amount over equal input intervals. Exponential functions multiply by the same factor over equal input intervals.

Fast decision rule

Compare differences for linear behavior and ratios for exponential behavior.

Worked steps

If outputs rise by $4$ for every one-unit input step, the pattern is linear. If outputs double for every step, it is exponential.

Common mistake

Using a straight line just because the data increase.

Mini check

Which pattern is exponential: equal differences or equal ratios?

Learning Objectives

Key idea

2.2.A — Construct functions of the real numbers that are comparable to arithmetic and geometric sequences. 2.2.B — Describe similarities and differences between linear and exponential functions.

Fast decision rule

Match the prompt verb to the representation it asks you to build or interpret.

Worked steps

$1$ . Name the given quantities. $2$ . Choose the matching function or representation. $3$ . Show the parameter or operation that changes. $4$ . Check one value in context.

Common mistake

Starting calculations before identifying what the input and output mean.

Mini check

Can you state what the input, output, and requested evidence are before choosing a formula?

Key Concepts

Key idea

A linear model has constant additive change; an exponential model has constant relative change. Both can be written for real inputs.

Fast decision rule

Ask “same amount?” for linear and “same percent or factor?” for exponential.

Worked steps

For a starting value $80$ that grows by $10$ each hour, use a linear model. For $80$ that grows by $10\%$ each hour, use an exponential model.

Common mistake

Treating $10\%$ as an additive increase of $10$ units.

Mini check

What does a constant ratio tell you about the model?

Relevant Representations

Key idea

Tables, graphs, and equations reveal different evidence about additive versus multiplicative change.

Fast decision rule

Inspect equal-length intervals; never compare raw changes across unequal intervals.

Linear or exponential?
Evidence	Linear	Exponential
Equal input steps	equal output differences	equal output ratios
Graph	constant slope	changing slope
Equation	$m x+b$	$a b^{x}$
Context	same amount per interval	same percent per interval

Worked steps

Compute the first differences and ratios in a table, then check whether the graph has constant slope or changing slope.

Common mistake

Comparing ratios when one output is zero or when intervals have different widths.

Mini check

Which graph feature separates a line from an exponential curve?

Essential Relationships or Formulas

Key idea

Use an exact relationship to expose the pattern before substituting values.

f(x)=mx+b

g(x)=ab^x

Fast decision rule

Keep the variables and restrictions visible, then substitute with units attached.

Worked example

Write the relationship, identify each parameter, substitute the known values, and check one output.

Common mistake

Using a familiar formula without checking its domain, base, interval, or unit assumptions.

Mini check

Which symbol or parameter carries the feature the prompt asks you to find?

Worked Examples

Key idea

A short, labeled chain of steps is easier to check than a page of unexplained algebra.

Fast decision rule

Write the representation first, calculate second, and interpret last.

Worked steps

Build one table of first differences and ratios, then choose the model whose pattern stays stable.

Common mistake

Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.

Mini check

Which step would you show so another student could reproduce the result?

Common Mistakes

Key idea

Most errors come from a mismatch between the pattern, the operation, and the allowed input.

Fast decision rule

Before submitting, check pattern, operation, domain, and units in that order.

Worked steps

Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.

Common mistake

Trusting a calculator output that hides a wrong model, sign, base, or domain.

Mini check

What independent check would catch your most likely error?

AP Exam Tips

Key idea

Use the topic workflow to connect the representation, the equation, and the context before calculating.

Fast decision rule

Read the prompt, choose the smallest reliable representation, and verify the result in context.

Worked steps

$1$ . Identify the pattern. $2$ . Write the matching form. $3$ . Show the key substitution or transformation. $4$ . Check units, domain, and direction.

Common mistake

Submitting a number without explaining why the selected model or transformation fits.

Mini check

Can you name the evidence that supports your final model or answer?

Practice

Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.

TRY IT IN A NEW SETTING

Can you transfer the method?

Two quantities start at $80$ . One increases by $10$ per hour; the other increases by $10\%$ per hour. Write both models for $t$ hours.

Compare your reasoning

The additive model is $L \left(t\right)=80+10 t$ . The proportional model is $E \left(t\right)=80 \left(1.1\right)^{t}$ . Their first-hour increases differ: $10$ units versus $8$ units.

Use it in a real question

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

Practise this topic Check the whole unit

2.2 · Change in Linear and Exponential Functions

Topic 2.2 · Change in Linear and Exponential 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?