Constructing Sinusoidal Function Models (College Board AP® Precalculus): Revision Note

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

    • $\text{period}=\frac{1}{\text{frequency}}$

  • 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

    • $\text{amplitude}=\left|a\right|=\frac{\text{max}-\text{min}}{2}$

  • and the vertical shift

    • $\text{vertical shift}=d=\frac{\text{max}+\text{min}}{2}$

• 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 $35°\text{F}$ to $85°\text{F}$

  • then the amplitude is $\left|a\right|=\frac{85-35}{2}=25$

  • 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 $\sin \theta$ crosses the midline going upward at $\theta =0$

• 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 $\cos \theta$ has a maximum point at $\theta =0$

• There can be more than one possible correct value for a phase shift

  • Adding or subtracting $2\pi$ to a phase shift determined by the methods above will give another valid phase shift value

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 $a\sin \left(bx+c\right)+d$

    • I.e., not in 'factored form' like $a\sin \left(b\right(\theta +c\left)\right)+d$

      • 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 $1\le t\le 365$

  • Or a model for the height of a point on a spinning wheel might only be valid for $t\ge 0$

• 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 $T\left(t\right)$, 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 $P$)

• Students must assign coordinates $(t,h(t\left)\right)$ 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 $\mathit{h}\mathbf{\left(}\mathit{t}\mathbf{\right)}$-coordinates (vertical)

    • Use the context to determine the maximum, minimum, and midline values

  • For $\mathit{t}$-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