AP®MathsCollege BoardPrecalculusStudy GuidesTrigonometric & Polar FunctionsSinusoidal Functions & ModelingConstructing Sinusoidal Function Models

Constructing Sinusoidal Function Models (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

Sinusoidal models

How can the period be determined from a context or data?

The period is
- the smallest interval of input values
- over which the output values complete one full cycle
  - and then begin to repeat
In practical terms, the period can be found by identifying the distance between consecutive maxima or consecutive minima
- E.g. if a sinusoidal function has maxima at $t = 5$ and $t = 17$
  - the period is $17 - 5 = 12$
The period can also be estimated from a table of values
- by looking for where the output values start cycling through the same pattern again
If the frequency is given (the number of complete cycles per unit)
- then remember that the period is its reciprocal
  - $period = \frac{1}{frequency}$
- E.g. if a quantity oscillates 4 times per second, the period is $\frac{1}{4}$ seconds

How can the amplitude and vertical shift be determined?

The maximum and minimum output values can be used to find both
- the amplitude
  - $amplitude = | a | = \frac{max - min}{2}$
- and the vertical shift
  - $vertical shift = d = \frac{max + min}{2}$

In a context-based question, the maximum and minimum are often stated directly
- or can be deduced from the given information
E.g. if a temperature ranges from $35 ° F$ to $85 ° F$
- then the amplitude is $|a| = \frac{85 - 35}{2} = 25$
- and the vertical shift is $d = \frac{35 + 85}{2} = 60$

How can the phase shift be determined?

Once the amplitude, vertical shift, and period are known, the phase shift can be estimated
- by comparing an actual known input-output pair
- to the values predicted by the model without any phase shift
In practice, the phase shift is determined by identifying where in the cycle the function starts
If a sine model is being used
- Identify the input value where the function first crosses the midline going upward
  - the phase shift equals that input value
- This works because the untransformed function $\sin θ$ crosses the midline going upward at $θ = 0$
If a cosine model is being used
- Identify the input value where the first maximum occurs
  - the phase shift equals that input value
- This works because the untransformed function $\cos θ$ has a maximum point at $θ = 0$
There can be more than one possible correct value for a phase shift
- Adding or subtracting $2 π$ to a phase shift determined by the methods above will give another valid phase shift value

Examiner Tips and Tricks

Remember that in a sinusoidal function of form $f (θ) = a \sin (b (θ + c)) + d$ or $g (θ) = a \cos (b (θ + c)) + d$ , the phase shift is equal to $- c$ (not $c$ ).

E.g. if a sine function first crosses the midline going upward at $θ = \frac{π}{3}$
- That is a shift of the basic sine function by $\frac{π}{3}$ to the right
- So $- c = \frac{π}{3} \Rightarrow c = - \frac{π}{3}$
- Therefore the function will be of the form $f (θ) = a \sin (b (θ - \frac{π}{3})) + d$

How can sinusoidal models be constructed using technology?

For a data set that appears to follow a sinusoidal pattern a sinusoidal regression can be performed using a graphing calculator
- This produces values for the parameters $a$ , $b$ , $c$ , and $d$ that best fit the data
- However be careful
  - Your calculator may give those values for a function in the form $a \sin (b x + c) + d$
  - I.e., not in 'factored form' like $a \sin (b (θ + c)) + d$
    - You can convert to factored form if necessary
Alternatively, the key values (amplitude, period, vertical shift) can be estimated from a graph or table
- and then used to build the model manually
Both approaches are valid on the exam
Sinusoidal regressions are especially useful when the data does not fall exactly on a sinusoidal curve

What is a contextual domain, and how does it affect the model?

Sinusoidal function models often have a contextual domain
- i.e. a restricted range of input values over which the model is meaningful
E.g. a model for daily temperature over a year might only be valid for $1 \leq t \leq 365$
- Or a model for the height of a point on a spinning wheel might only be valid for $t \geq 0$
Within the contextual domain, the model can be used
- to predict values of the dependent variable
- from values of the independent variable
The model can also be used in reverse
- i.e. given a value of the dependent variable
- the independent variable can be predicted
  - by solving the equation for the independent variable
- E.g. given a temperature model $T (t)$ , you could be asked to find at what time $t$ the temperature reaches a particular value

Outside the contextual domain, the model's predictions may not be meaningful
- even though the mathematical function continues to produce output values

How should the five key points on a sinusoidal graph be labeled?

On the exam, free response question 3 always presents
- a generic sinusoidal graph (without scale or axes)
- showing two full cycles
- with five labeled points (usually $F$ , $G$ , $J$ , $K$ , and $P$ )

Graph of a sine wave with three peaks and two troughs, labelled points F, G, J, K, P on lines; dashed and solid horizontal lines intersect the curve.

Students must assign coordinates $(t, h (t))$ to each point based on the context
The five points correspond to the key features of the sinusoidal graph
- $F$ and $P$ are two successive maxima
- $J$ is a minimum
- $G$ and $K$ are on the midline
  - $G$ while the function is decreasing, $K$ while it is increasing

To find the coordinates
- For $h (t)$ -coordinates (vertical)
  - Use the context to determine the maximum, minimum, and midline values
- For $t$ -coordinates (horizontal)
  - Use the period and the phase shift to determine the time at each key feature
  - Horizontally, the key points are all one quarter of the period apart
    - i.e. from max to midline is one quarter period
    - as are midline to min, min to midline, and midline to max

Examiner Tips and Tricks

For a question of this sort, the $h (t)$ -coordinates are generally easier to determine than the $t$ -coordinates. The chief reader reports consistently show that most students who earn the $h (t)$ -coordinate point go on to struggle more with the $t$ -coordinates.

Make sure you understand how the period relates to the spacing between key points.

The time between any two adjacent key points (max → midline → min → midline → max) is always one quarter of the period

Also note that the $t$ -coordinates do not have to start at $t = 0$ .

The context determines where the cycle begins, so pay close attention to what the context tells you about the starting conditions

Worked Example

A weight attached to a spring oscillates vertically. At time $t = 3$ seconds, the weight is at its highest point, 14 cm above the table. It then moves downward to its lowest point, 4 cm above the table, and returns to 14 cm above. One complete oscillation takes $10$ seconds.

The sinusoidal function $h$ models the height of the weight above the table, in cm, as a function of time $t$ , in seconds.

The graph of $h$ and its dashed midline for two full cycles is shown. Five points, $F$ , $G$ , $J$ , $K$ , and $P$ , are labeled on the graph. No scale is indicated, and no axes are presented.

Determine possible coordinates $(t, h (t))$ for the five points: $F$ , $G$ , $J$ , $K$ , and $P$ .

Answer:

First find the $h (t)$ -coordinates

From the context, the maximum height is 14 cm and the minimum height is 4 cm

This gives

$midline = \frac{14 + 4}{2} = 9$

Therefore

$F$ and $P$ (maxima) have $h (t) = 14$

$G$ and $K$ (midline) have $h (t) = 9$

$J$ (minimum) has $h (t) = 4$

Now find the $t$ -coordinates:

The period (time for one complete oscillation) is $10$ seconds

so the time between consecutive key points is

$\frac{10}{4} = 2.5$ seconds

The weight is at its highest point at $t = 3$ , which corresponds to $F$

Therefore

$F$ : $t = 3$ (maximum)

$G$ : $t = 3 + 2.5 = 5.5$ (midline, going down)

$J$ : $t = 5.5 + 2.5 = 8$ (minimum)

$K$ : $t = 8 + 2.5 = 10.5$ (midline, going up)

$P$ : $t = 10.5 + 2.5 = 13$ (next maximum)

So the five points are

$F (3, 14) G (5.5, 9) J (8, 4) K (10.5, 9) P (13, 14)$

Worked Example

The figure shows the graph of a sinusoidal function $f$ , along with a dashed line showing the midline of the function. The function $f$ can be written in the form $f (x) = a \sin (b (x + c)) + d$ , where $b > 0$ . What is the value of $b$ ?

Graph of a sine wave with two peaks at (3,8) and (15,8), and two troughs at (-3, -4) and (9, -4) on a grid. The x and y axes marked. Dotted midline crosses the y-axis at 2.

(A) $\frac{π}{12}$

(B) $\frac{π}{6}$

(D) $12$

Answer

The value of $b$ is connected to the period of the function

The period can be determined from the distance between two consecutive maxima

and the maxima on the graph are at $(3, 8)$ and $(15, 8)$ , so

$period = 15 - 3 = 12$

Now use the relationship $period = \frac{2 π}{| b |}$ :

$\begin{array}{rcl} 12 & = & \frac{2 π}{|b|} \\ |b| & = & \frac{2 π}{12} \\ |b| & = & \frac{π}{6} \end{array}$

And you are told that $b > 0$ , so $b = \frac{π}{6}$

(B) $\frac{π}{6}$

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Constructing Sinusoidal Function Models (College Board AP® Precalculus): Study Guide

Sinusoidal models

How can the period be determined from a context or data?

How can the amplitude and vertical shift be determined?

How can the phase shift be determined?

Examiner Tips and Tricks

How can sinusoidal models be constructed using technology?

What is a contextual domain, and how does it affect the model?

How should the five key points on a sinusoidal graph be labeled?

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis