Knowledge Notes | Carey AP Precalculus Practice Lab

THINK BEFORE YOU CALCULATE

A height keeps increasing. Can its rate of change still decrease?

During the first three seconds, a ball has height $h(t)=-t^2+6t$ metres. Its heights at $t=0,1,2,3$ seconds are $0,5,8,9$ metres.

Make a prediction. Keep it in mind as you test the representation.

For the specified quadratic, height increases while the rate decreases; finite samples alone do not establish continuous concavity.

Loading the interactive investigation…

FROM OBSERVATION TO EVIDENCE

Build an answer that explains the mathematics

1
Track the outputs
Read the ordered pairs as time and height, in that order. All four heights are different, and each height after the first is larger.
Write the evidence
As time increases from $0$ to $3$ seconds, the height increases from $0$ to $9$ metres.
2
Compare equal intervals
The input intervals are all one second long. Compute output change divided by input change on each interval.
Write the evidence
$\frac{5-0}{1-0}=5,\qquad\frac{8-5}{2-1}=3,\qquad\frac{9-8}{3-2}=1$ metres per second.
3
Connect the two claims
All three average rates are positive, but they get smaller. For this displayed quadratic, that agrees with an increasing, concave-down graph on the given interval.
Write the evidence
The ball is rising at a decreasing rate. Increasing describes height; decreasing describes the rates of change.

Read one complete model response

Over $0\le t\le3$ , the height is increasing. The average rates on successive one-second intervals are $5$ , $3$ , and $1$ metres per second, so the height increases at a decreasing rate; the displayed quadratic is concave down.

Carey teaching response. Use the evidence requested by your actual question; these steps are not additional official scoring points.

A COMMON WRONG TURN

Spot the exact reasoning error

Incomplete or incorrect response

“The graph is decreasing because the slopes are decreasing.”

What needs to change

The slopes are still positive. Decreasing positive slopes mean the height rises more slowly, not that the height falls. A few table values alone do not prove behavior between every observation unless a function model or other information supports it.

TRY THE IDEA IN A NEW SETTING

Can you transfer the method?

A tank contains $12,19,24,27$ litres at $0,1,2,3$ minutes. Describe the observed output change and the successive average rates. Do these four measurements alone prove continuous concavity?

Compare your reasoning

The recorded volumes increase. The average rates are $7,5,3$ litres per minute, so the sampled rates decrease. The four measurements alone do not establish what happens between them or prove concavity of an unspecified continuous function.

CONNECT THE WHOLE TOPIC

Complete topic reference

Use the guided example above, then connect it to the remaining ideas and representations in this topic.

What You Need to Know

Big idea

A function pairs each allowed input with one output. Covariation describes how the output responds as the input changes.

Read it in order

Name the input, name the output, state the interval, then describe whether the output increases, decreases, or stays constant.

Worked example

If time increases from $0$ to $3$ hours while distance changes from $0$ to $5$ miles, distance and time vary together. The data show the relationship; they do not by themselves prove a constant rate.

Watch the distinction

An output value is one snapshot. Increasing or decreasing describes a comparison across ordered inputs.

Mini check

In a table of time and water depth, which quantity is the input? What sentence describes a depth that falls while time increases?

Learning Objectives

1.1.A

Describe how input and output values vary together by comparing function values.

Evidence for 1.1.A

State the input and output, compare the values in the requested interval, and use precise words such as increasing, decreasing, or constant.

1.1.B

Construct a graph that represents two quantities changing in a contextual situation.

Evidence for 1.1.B

Place the correct starting value, match the direction of change, preserve the domain, and mark any turning or flat interval supported by the context.

Mini check

Could a graph have the right overall direction but still be wrong? Name one detail that would make it invalid.

Key Concepts

Function notation

The input $a$ produces the output $f(a)$ . The domain lists allowed inputs; the range lists the outputs that occur.

f(a)\text{ is the output paired with input }a.

Behavior

A graph is increasing on an interval when larger inputs give larger outputs. It is decreasing when larger inputs give smaller outputs, and constant when the output does not change.

Image and preimage

The image of an input is its output. A preimage is an input that produces a selected output. Keep the direction of the pairing clear.

Curvature is separate

A function can be increasing while its increases get smaller, or decreasing while its decreases get larger. Direction and changing rate are different observations.

Mini check

If a graph is flat from $x=2$ to $x=5$ , what happens to the output? Is the function increasing on that interval?

Relevant Representations

Use the representation that exposes the evidence the question asks for: words explain the context, a table pairs values, a graph shows behavior over an interval, and notation names an exact output.

Vector teaching card that connects input and output variables to function notation, paired table values, and an exact contextual graph.

Choose the representation that answers the question
Representation	Look for	Use it to
Words	which quantity changes first and what is allowed	identify variables, units, and domain
Table	paired values and repeated comparisons	compare function values
Graph	direction, flat intervals, turning points, endpoints	describe behavior and build a model
Notation	an input-output pairing	evaluate or name an image

Mini check

Which representation would you inspect first to find a flat interval: a table, a graph, or an equation? Explain what evidence you would use.

Essential Relationships or Formulas

A change compares two values. An average rate compares the output change with the input change, so the denominator is a change in input, not the second input value.

\Delta y=f(b)-f(a)

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

Vector formula card showing average rate of change as output change divided by input change, with a secant line joining two points.

Interpret the sign

A positive average rate means the output rose over the interval; a negative rate means it fell. Always attach units such as miles per hour or meters per minute.

Mini check

If $f(4)=10$ and $f(9)=2$ , find the average rate from $x=4$ to $x=9$ . Include the sign and units.

Worked Examples

Example

A runner records distance $d(t)$ in miles at $t=0,1,2,3$ hours as $0$ , $2$ , $5$ , $5$ .

Runner data
$t$ (hours)	$0$	$1$	$2$	$3$
$d(t)$ (miles)	$0$	$2$	$5$	$5$

Step 1

From $t=0$ to $t=2$ , distance increases from $0$ to $5$ miles. Step $2$ — From $t=2$ to $t=3$ , distance is constant at $5$ miles. Step $3$ — A faithful graph starts at $(0,0)$ , rises through the first values, and is horizontal from $t=2$ to $t=3$ .

Runner distance over time

Check

The domain is $0\le t\le 3$ . Do not extend the line beyond $t=3$ unless the context gives information after the recorded interval.

Common Mistakes

Mistake 1

Calling a function increasing because its final value is larger. Check every interval the question names; a graph may rise, flatten, and fall.

Mistake 2

Confusing a negative change with a negative output. An output can be positive while its change is negative.

Mistake 3

Ignoring the context. A graph with the wrong starting value, endpoint, unit, or domain is not a valid model even if its shape looks reasonable.

Vector graph vocabulary card showing increasing, decreasing, constant, increasing while slowing, and decreasing while steepening.

Repair routine

Circle the input, underline the output, mark the interval, and decide whether the question asks for a value, change, rate, or graph.

Mini check

A graph rises overall but has one downward interval. Is it increasing on the entire domain? Explain using intervals.

AP Exam Tips

Use an evidence sentence

As the input increases from ___ to ___, the output ___. This forces the interval, direction, and variables into your answer.

Graph-choice scan

Check the starting point, direction on each interval, flat pieces, turning points, endpoint restrictions, and units in that order.

When a rate is requested

Write the output difference first, write the input difference second, preserve endpoint order, and report the units.

Final check

Point to one table entry, graph feature, or function value that proves your choice. “It looks right” is not mathematical evidence.

Take the next step

Use an independent practice question to test what you can now explain.

Practise change and covariation Connect to rates of change

1.1 · Change in Tandem

Topic 1.1 · Change in Tandem

A height keeps increasing. Can its rate of change still decrease?

Build an answer that explains the mathematics

Track the outputs

Compare equal intervals

Connect the two claims

Spot the exact reasoning error

Can you transfer the method?

Complete topic reference