Knowledge Notes | Carey AP Precalculus Practice Lab

For a positive starting value, read the multiplier separately from the initial value.

What a strong response shows

Read the value at input zero separately from the base.
For positive initial value, compare the base with $1$ .
Convert a decay factor to the percent lost, not the percent retained.

Loading the interactive investigation…

What You Need to Know

Key idea

In the positive-output exponential models studied here, a positive starting value is multiplied by a positive base raised to the input. Exponential functions may also be reflected by a negative nonzero outside scale; do not apply the positive-output growth/decay wording to that case without checking it.

Fast decision rule

Read the starting value from the y-intercept, then classify the base: above $1$ means growth; between $0$ and $1$ means decay.

Worked steps

For a model with starting value $240$ and factor $1.08$ , the output grows $8\%$ per input unit.

Common mistake

Calling a factor of $0.92$ growth because the number $92$ is large.

Mini check

What happens when the base is between $0$ and $1$ ?

Learning Objectives

Key idea

2.3.A — Identify key characteristics of 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

Key features include initial value, base, growth or decay, domain, range, intercept, and horizontal asymptote.

Fast decision rule

Separate the multiplier outside the power from the base inside the power.

Worked steps

In an unshifted model, the initial value is the y-intercept and the horizontal asymptote is $y=0$ .

Common mistake

Confusing the initial value with the growth factor.

Mini check

Which parameter controls the percent change?

Relevant Representations

Key idea

The equation, table, graph, and context should tell the same growth or decay story.

Fast decision rule

Check the y-intercept, one interval factor, and long-run direction across representations.

Read an exponential model
Feature	Where to find it	Meaning
Initial value	coefficient a	output at input $0$
Base	factor $b$	multiplier per unit
Asymptote	long-run graph level	value approached
Domain/range	context and graph	allowed inputs and outputs

Worked steps

Use two table rows to find the factor, then confirm that the graph approaches the correct asymptote.

Common mistake

Extending an exponential model into negative outputs when the context requires positive quantities.

Mini check

What visual evidence shows decay?

Essential Relationships or Formulas

Key idea

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

f(x)=ab^x,\qquad a>0,\ b>0,\ b\ne1

b>1\Rightarrow\text{growth},\qquad 0<b<1\Rightarrow\text{decay}

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

Identify the initial value and base, classify growth or decay, and state the asymptote before evaluating a later input.

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?

Describe $P(t)=120(1.08)^t$ when $t$ is measured in years.

Compare your reasoning

The initial amount is $120$ , and each one-year increase multiplies it by $1.08$ , representing $8\%$ annual growth. For this unshifted positive exponential, the horizontal asymptote is $y=0$ .

Use it in a real question

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

Practise this topic Check the whole unit

2.3 · Exponential Functions

Topic 2.3 · 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?