Knowledge Notes | Carey AP Precalculus Practice Lab

THINK BEFORE YOU CALCULATE

How can one complete revolution determine an entire height model?

A passenger reaches a maximum height of $26$ metres and a minimum height of $14$ metres. The wheel turns at constant angular speed; one revolution takes $12$ minutes. At $t=0$ , the passenger is at the highest point and then moves down.

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

The same passenger moves from maximum through a descending midline, minimum and ascending midline, then returns to the maximum. — For a wheel turning at constant angular speed, a complete revolution produces one sinusoidal height cycle. Amplitude is the radius, the midline is the centre height, and period is the time for one full revolution. The artwork is conceptual; use the exact model and units in the question.

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FROM OBSERVATION TO EVIDENCE

Build an answer that explains the mathematics

1
Find the vertical scale
The amplitude is half the total height range. The midline is halfway between the extreme heights.
Write the evidence
$A=\frac{26-14}{2}=6\text{ m},\qquad d=\frac{26+14}{2}=20\text{ m}.$
2
Place five quarter-cycle anchors
One quarter-period is three minutes. Starting at the maximum, move down through the midline, to the minimum, up through the midline and back to the maximum.
Write the evidence
$(0,26),\ (3,20),\ (6,14),\ (9,20),\ (12,26).$
3
Build and verify an exact model
A positive cosine starts at a maximum. Its angular frequency is a full radian turn divided by the period. Check both the value and nearby motion.
Write the evidence
$b=\frac{2\pi}{12}=\frac{\pi}{6},\qquad h(t)=20+6\cos\!\left(\frac{\pi t}{6}\right).$ At $t=0$ , the height is $26$ metres; just afterward it decreases.

Read one complete model response

The amplitude is $6$ metres, the midline is $h=20$ metres and the period is $12$ minutes. One model is $h(t)=20+6\cos(\pi t/6)$ . It starts at $26$ metres and decreases immediately afterward, matching the stated starting position and motion.

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 period is $12$ , so the model is $20+6\cos(12t)$ .”

What needs to change

The coefficient of time is angular frequency, not the period. For cosine in radians, use $b=2\pi/T$ . A correct maximum at the start does not by itself verify the cycle length.

TRY THE IDEA IN A NEW SETTING

Can you transfer the method?

A second wheel also turns at constant angular speed. It has maximum $30$ metres, minimum $10$ metres and period $8$ minutes. At $t=0$ , its passenger crosses the midline moving upward. Give one model.

Compare your reasoning

The amplitude is $10$ metres, midline $20$ metres and angular frequency $\pi/4$ radians per minute. One model is $h(t)=20+10\sin(\pi t/4)$ . Its starting height is $20$ metres and it initially increases.

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

Key idea

A sinusoidal model describes data that repeats around a central level with a regular cycle.

Fast decision rule

Check repetition, range, and timing before choosing a sinusoidal model.

Worked steps

Plot data, estimate max and min, estimate cycle length, and use those features to build a model.

Common mistake

Forcing a sinusoid onto data with no stable repeat.

Mini check

What evidence makes a sinusoidal model plausible?

Learning Objectives

Key idea

3.7.A — Construct sinusoidal function models of periodic phenomena.

Fast decision rule

For a modeling prompt, state what each parameter means in the context.

Worked steps

$1$ . Identify input and output units. $2$ . Estimate midline and amplitude. $3$ . Estimate period. $4$ . Choose phase from a clear event. $5$ . Validate with data.

Common mistake

Reporting a parameter without its contextual unit.

Mini check

Which step tests whether the model is useful?

Key Concepts

Key idea

A model is judged by context, parameter meaning, and residual size—not by a smooth curve alone.

Fast decision rule

Interpret first, fit second, validate third.

Worked steps

Compare predicted and observed values at several inputs and look for a systematic pattern in the residuals.

Common mistake

Treating a good fit inside the data range as proof beyond the data range.

Mini check

What do residuals reveal about a model?

Relevant Representations

Key idea

A modeling workflow connects a scatter plot, parameter estimates, a formula, and predictions.

Modeling evidence
Evidence	Question
repeat timing	Is the cycle stable?
vertical range	What are max and min?
starting stage	Where does the cycle begin?
residual pattern	What does the model miss?

Fast decision rule

Keep units visible in every representation.

Worked steps

Use the graph to locate peaks and troughs, the table to check timing, and the formula to predict between observations.

Common mistake

Changing the input unit after estimating the period.

Mini check

Which representation is best for checking residuals?

Essential Relationships or Formulas

Key idea

The model parameters encode the center, vertical range, cycle length, and starting stage.

y=A\sin\bigl(B(x-h)\bigr)+D

|A|=\frac{y_{\max}-y_{\min}}{2}

P=\frac{2\pi}{|B|}

Fast decision rule

Use context units to check every parameter before predicting.

Worked steps

Convert a measured period into the input scale, then use an observed maximum or midline crossing to select the phase.

Common mistake

Using a period measured in days inside a model whose input is weeks.

Mini check

How can units expose a bad parameter?

Worked Examples

Key idea

A good model explains why the curve has its height, timing, and phase.

Fast decision rule

After fitting, test a known data point and one nearby prediction.

Worked steps

For weekly demand that peaks every four weeks, set the cycle length from the peak spacing, place the midline halfway between high and low demand, and compare predictions with observed weeks.

Common mistake

Choosing phase from an arbitrary first data point.

Mini check

Which observed event is most useful for phase?

Common Mistakes

Key idea

Modeling errors hide in units, phase, and overconfident predictions.

Fast decision rule

Check input unit, output unit, phase event, and data range.

Worked steps

Write a short sentence for each parameter and compare it with the original context.

Common mistake

Calling extrapolation interpolation.

Mini check

What is the difference between interpolation and extrapolation?

AP Exam Tips

Key idea

A model answer needs both a formula and a sentence interpreting its parameters.

Practice

Use the Questionbank set below to test this skill.

Fast decision rule

Give a prediction only inside a defensible context range unless the prompt asks otherwise.

Worked steps

State the model, substitute the requested input, and interpret the output with units and an accuracy caveat.

Common mistake

Reporting a decimal without the original unit.

Mini check

Does your prediction answer the contextual question?

Take the next step

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

Practise sinusoidal models Apply the method to official FRQ3

3.7 · Sinusoidal Function Context and Data Modeling

Topic 3.7 · Sinusoidal Function Context and Data Modeling

How can one complete revolution determine an entire height model?

Build an answer that explains the mathematics

Find the vertical scale

Place five quarter-cycle anchors

Build and verify an exact model

Spot the exact reasoning error

Can you transfer the method?

Complete topic reference