Define sinusoidal function .

A sinusoidal function is any function that can be obtained by applying additive and multiplicative transformations to the sine function. Both the sine function and the cosine function are themselves sinusoidal functions.

True or False? A sinusoidal function is increasing everywhere its graph is concave up.

False. Concavity and direction are independent: concavity says how the rate of change is behaving, while direction says whether the outputs are rising or falling. On a sinusoidal graph there are stretches that are concave up while the function is still decreasing.

Where on a sinusoidal graph is it concave up, and where concave down?

The graph is concave down on the sections around each maximum and concave up on the sections around each minimum. It alternates between the two as the input increases, changing over at the points of inflection, which all lie on the midline .

A sinusoidal function is increasing and its graph is concave down. What is happening to its rate of change?

The rate of change is decreasing . The function is still rising, but rising more and more slowly, which is what happens on the approach to a maximum.

True or False? A phase shift changes which input values give the maximum, but not what the maximum value is.

True. A horizontal translation slides the graph sideways only, so the maximum, the minimum, the amplitude and the midline all keep their values. What changes is where the maxima, the minima and the midline crossings occur.

Once an equation has been written for a sinusoidal graph, how can it be checked?

Substitute the coordinates of a known point from the graph and confirm that the equation produces them. Testing a maximum and one midline crossing will catch a wrong phase shift or a wrong period, which are the two parameters easiest to get wrong.

What distinguishes the two midline points within one cycle of a sinusoidal graph?

The direction the function is travelling as it crosses. One of them sits between a maximum and the following minimum, where the function is decreasing , and the other between a minimum and the following maximum, where it is increasing .

True or False? A sinusoidal model of a given set of data has only one correct phase shift.

False. Adding or subtracting a whole number of periods to the phase shift gives a different value that describes exactly the same graph. So the same data can be modeled by several equations differing only in that one parameter.

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Sinusoidal Functions & Modeling (College Board AP® Precalculus): Flashcards

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Define sinusoidal function.

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Cards in this collection (26)

Define sinusoidal function.
A sinusoidal function is any function that can be obtained by applying additive and multiplicative transformations to the sine function.
Both the sine function and the cosine function are themselves sinusoidal functions.
Why can any sinusoidal function be written using either sine or cosine?
Because cosine is itself a phase shift of sine:
$\cos θ = \sin (θ + \frac{π}{2})$
Absorbing that shift into the phase turns any transformed sine into a transformed cosine, and the same step run backwards does the reverse.
Fill in the two gaps about the period and frequency of a sinusoidal function.
The period and the frequency of a sinusoidal function are always $______$ of each other, so for the base functions $\sin θ$ and $\cos θ$ the period is $2 π$ while the frequency is $______$ instead.
The completed sentence is:
The period and the frequency of a sinusoidal function are always reciprocals of each other, so for the base functions $\sin θ$ and $\cos θ$ the period is $2 π$ while the frequency is $\frac{1}{2 π}$ instead.
Frequency counts how many complete cycles fit into one unit of input, so the two must multiply to $1$ .
How are the amplitude and the midline of a sinusoidal function found from its maximum and minimum?
The amplitude is half the difference between them, $\frac{maximum - minimum}{2}$ , which is also the distance from the midline up to the maximum.
The midline is the horizontal line at their average, $y = \frac{maximum + minimum}{2}$ .
For $\sin θ$ and $\cos θ$ themselves the amplitude is $1$ and the midline is $y = 0$ .
True or False?
A sinusoidal function is increasing everywhere its graph is concave up.
False.
Concavity and direction are independent: concavity says how the rate of change is behaving, while direction says whether the outputs are rising or falling.
On a sinusoidal graph there are stretches that are concave up while the function is still decreasing.
Where on a sinusoidal graph is it concave up, and where concave down?
The graph is concave down on the sections around each maximum and concave up on the sections around each minimum.
It alternates between the two as the input increases, changing over at the points of inflection, which all lie on the midline.
A sinusoidal function is increasing and its graph is concave down. What is happening to its rate of change?
The rate of change is decreasing.
The function is still rising, but rising more and more slowly, which is what happens on the approach to a maximum.
In $f (θ) = a \sin (b (θ + c)) + d$ each constant controls one feature. Fill in the two gaps.
The constant $a$ sets the $______$ of the function, while $b$ sets its $______$ instead.
The completed sentence is:
The constant $a$ sets the amplitude of the function, while $b$ sets its period instead.
The other two constants are the translations: $c$ produces a phase shift of $- c$ , and $d$ moves the midline to $y = d$ .
For $g (θ) = \sin θ + d$ what does the constant $d$ do to the graph?
It moves the midline from $y = 0$ to $y = d$ , so the function oscillates about that line rather than about the horizontal axis.
The maximum value becomes $1 + d$ and the minimum becomes $- 1 + d$ , while the amplitude and the period are unaffected.
For $g (θ) = \sin (θ + c)$ which way does the graph move, and by how much?
It moves horizontally by $- c$ , a translation known as a phase shift.
So a positive value of $c$ shifts the graph to the left by $c$ units, and a negative value shifts it to the right by $|c|$ units.
True or False?
A phase shift changes which input values give the maximum, but not what the maximum value is.
True.
A horizontal translation slides the graph sideways only, so the maximum, the minimum, the amplitude and the midline all keep their values.
What changes is where the maxima, the minima and the midline crossings occur.
For $g (θ) = a \sin θ$ what does the constant $a$ do?
It dilates the graph vertically, so the amplitude becomes $|a|$ .
With no vertical shift present the maximum value becomes $|a|$ and the minimum becomes $- |a|$ , while the period and the midline are unaffected.
For $g (θ) = \sin (b θ)$ how does the period change?
The graph is dilated horizontally by a factor of $\frac{1}{|b|}$ , so the new period is $\frac{2 π}{|b|}$ .
So $\sin (4 θ)$ has a period of $\frac{π}{2}$ , which fits four complete cycles into an interval of width $2 π$ .
True or False?
A negative value of $b$ in $\sin (b θ)$ gives a graph that cannot be produced with a positive value of $b$ .
False.
A negative $b$ reflects the graph over the vertical axis, and because of the symmetry of sine and cosine that reflection can always be expressed as a phase shift instead.
So every sinusoidal graph can be written with a positive value of $b$ .
A sinusoidal graph has consecutive maxima at $x = 1$ and at $x = 5$ . What is the value of $b$ ?
The period is the horizontal distance between them, so the period is $5 - 1 = 4$ .
Then rearranging the period formula gives $|b| = \frac{2 π}{period} = \frac{2 π}{4} = \frac{π}{2}$ .
A sinusoidal graph has a maximum of $7$ and a minimum of $1$ . Fill in the two gaps.
Its amplitude is $______$ and its midline is the line $y =______$ as a result.
The completed sentence is:
Its amplitude is $3$ and its midline is the line $y = 4$ as a result.
Half the difference of $7$ and $1$ is $3$ , and the average of $7$ and $1$ is $4$ .
For a cosine model, which feature of the graph gives the phase shift?
The first maximum.
The base function $\cos θ$ has its maximum at $θ = 0$ , so if the graph's first maximum sits at $θ = h$ then the graph has been shifted $h$ units to the right.
For a sine model, which feature of the graph gives the phase shift?
The point where the graph first crosses the midline going upward.
The base function $\sin θ$ crosses its midline going upward at $θ = 0$ , so if the graph does so at $θ = h$ then it has been shifted $h$ units to the right.
True or False?
In $a \cos (b (θ + c)) + d$ , the value of $c$ is the distance the graph has been shifted.
False.
The shift is $- c$ rather than $c$ , so the two differ in sign.
A graph whose first maximum sits at $θ = \frac{π}{3}$ has been shifted $\frac{π}{3}$ to the right, and its equation therefore contains $c = - \frac{π}{3}$ .
Once an equation has been written for a sinusoidal graph, how can it be checked?
Substitute the coordinates of a known point from the graph and confirm that the equation produces them.
Testing a maximum and one midline crossing will catch a wrong phase shift or a wrong period, which are the two parameters easiest to get wrong.
For a sinusoidal function whose midline is $y = d$ what does a negative value of $a$ do to the graph?
It reflects the graph over its midline, so the peaks and the troughs swap places.
The maximum and minimum values themselves are unchanged, because the graph still reaches $|a|$ above and $|a|$ below the line $y = d$ .
A quantity reaches its maximum at $t = 5$ and again at $t = 17$ . What is the period of a sinusoidal model for it?
The period is $17 - 5 = 12$ , the input-value distance between consecutive maxima.
Two consecutive minima would give the same answer, and so would any two consecutive corresponding points on the graph.
A quantity oscillates $4$ times per second. What is the period of a sinusoidal model for it?
The period is $\frac{1}{4}$ of a second.
The figure of $4$ cycles per second is the frequency, and the period is its reciprocal.
A sinusoidal model has a period of $10$ seconds and reaches a maximum at $t = 3$ . Fill in the two gaps.
Its next crossing of the midline is at $t =______$ and the minimum after that is at $t =______$ seconds.
The completed sentence is:
Its next crossing of the midline is at $t = 5.5$ and the minimum after that is at $t = 8$ seconds.
Successive key points are each one quarter of a period apart, and a quarter of $10$ is $2.5$ .
What distinguishes the two midline points within one cycle of a sinusoidal graph?
The direction the function is travelling as it crosses.
One of them sits between a maximum and the following minimum, where the function is decreasing, and the other between a minimum and the following maximum, where it is increasing.
True or False?
A sinusoidal model of a given set of data has only one correct phase shift.
False.
Adding or subtracting a whole number of periods to the phase shift gives a different value that describes exactly the same graph.
So the same data can be modeled by several equations differing only in that one parameter.

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