Knowledge Notes | Carey AP Precalculus Practice Lab

Use the same endpoint order; this is an average rate over the whole interval.

What a strong response shows

Subtract outputs in the same order as inputs.
Divide by the interval width, even when it is not $1$ .
Distinguish a secant rate from a value at one input.

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What You Need to Know

Key idea

A rate of change compares how much an output changes with how much the input changes.

Fast decision rule

Name both quantities and the interval before calculating. Keep the endpoint order the same everywhere.

Worked example

A height change of $18$ feet over $6$ seconds means $3$ feet per second. The units tell you what the rate describes.

Common mistake

Using the output value instead of the output change, or reporting a rate without its units.

Mini check

A quantity drops by $12$ units while the input increases by $4$ . Is the rate positive or negative?

Learning Objectives

Key idea

1.2.A compares rates near two points; 1.2.B describes how two quantities vary at different points or intervals.

Fast decision rule

If the question says compare, compute or read both rates first; if it says describe, state direction, interval, and units.

Worked steps

$1$ . Mark the two intervals. $2$ . Find each output change. $3$ . Divide by each input change. $4$ . Compare the signs and sizes.

Common mistake

Comparing raw output changes when the intervals have different widths.

Mini check

What extra information is needed before two output changes can be compared fairly?

Key Concepts

Key idea

Average rate describes an interval; a nearby interval can show how behavior changes around a point.

Fast decision rule

A steeper positive rate means faster increase; a less positive or negative rate means the output is increasing more slowly or decreasing.

Worked example

Two intervals can both rise while the second has a smaller rate. The function is still increasing, but its increase is slowing.

Common mistake

Calling the function decreasing because its rate became smaller while the rate is still positive.

Mini check

If the rates are $8$ and $3$ on two consecutive intervals, what changed: direction, speed of increase, or both?

Relevant Representations

Key idea

Context, table, graph, and formula show the same rate evidence in different ways.

Fast decision rule

Use a table for exact changes, a graph for interval steepness, and a formula for an exact numerical comparison.

Choose the representation that exposes the rate
Representation	Look for	Use it to
Context	quantities and units	name input and output
Table	paired values and intervals	calculate changes
Graph	secant steepness	compare interval behavior
Formula	two endpoint values	compute a rate

Common mistake

Reading a graph’s height as its rate.

Mini check

Which representation makes units easiest to keep visible?

Essential Relationships or Formulas

Key idea

Average rate is output change divided by input change.

\operatorname{AROC}_{[a,b]}=\frac{f(b)-f(a)}{b-a}

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

Fast decision rule

Substitute the later endpoint first in both numerator and denominator, then simplify with units.

Worked example

For an output of $10$ at the first input and $22$ at the second input, the output change is $12$ . If the input change is $4$ , the average rate is $3$ output units per input unit.

Common mistake

Reversing only one difference, which changes the sign.

Mini check

What must happen to the sign of the rate when the output falls while the input rises?

Worked Examples

Key idea

A reliable solution turns a story or graph into two endpoint values and one interval.

Fast decision rule

Write the interval, compute the two changes, divide, and interpret the result in context.

Worked steps

$1$ . Label the endpoints a and $b$ . $2$ . Read the two endpoint outputs. $3$ . Calculate output change and input change. $4$ . Divide. $5$ . State the units and direction.

Worked example

If a graph moves from $\left(2,\,7\right)$ to $\left(5,\,16\right)$ , the average rate is $3$ output units per input unit. The positive sign means the output rises over that interval.

Common mistake

Skipping the interval and using a point value as if it were a rate.

Mini check

What five pieces should appear in a complete rate statement?

Common Mistakes

Key idea

Most errors are endpoint, sign, interval, or unit errors.

Fast decision rule

Circle the two inputs, underline their outputs, and check the denominator before selecting an answer.

Worked example

A drop from $18^{\circ} C$ to $12^{\circ} C$ is a change of $-6^{\circ} C$ . A rate requires the time interval as well.

Common mistake

Confusing a negative change with a negative output.

Mini check

Can a positive output have a negative rate? Explain with one sentence.

AP Exam Tips

Key idea

AP questions often hide the rate in a table, graph, or context sentence.

Fast decision rule

Read the requested interval first, then compare changes, signs, and units in that order.

Worked steps

$1$ . Identify input and output. $2$ . Lock the interval. $3$ . Compute or read both rates. $4$ . Eliminate choices with the wrong sign or units. $5$ . State what the rate means.

Common mistake

Choosing the interval with the biggest vertical change without dividing by its horizontal width.

Mini check

Before submitting, can you point to the exact numerator and denominator?

Practice

Use the Questionbank items below to compare interval rates and explain how two quantities vary.

TRY IT IN A NEW SETTING

Can you transfer the method?

A quantity is $11$ at time $2$ minutes and $23$ at time $5$ minutes. Calculate and interpret its average rate.

Compare your reasoning

The average rate is $\frac{23-11}{5-2}=4$ quantity units per minute. The endpoint output increases by $4$ units per minute on average; this does not prove a constant rate throughout.

Use it in a real question

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

Practise this topic Check the whole unit

1.2 · Rates of Change

Topic 1.2 · Rates of Change

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?