Unit 3 · Topic 3.7

Sinusoidal Function Context and Data Modeling

To model a periodic context, get the midline and amplitude from the greatest and least values, the period (and b = 2π/period ) from the cycle length or frequency, and the shift from one landmark time.

Practice 1: Procedural and Symbolic FluencyPractice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

A harbor master needs to know when the water will be deep enough for a ship. A greenhouse needs its vents to open before the afternoon peak. Neither has a formula handed to them, only a few facts: the highest and lowest values, when they happen, and how long a cycle takes. That is enough to build a sinusoidal model and answer the question, and it is exactly the task of the periodic modeling question on the exam.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A sinusoid has period 12. In $f(t)$ $= a\cos(b(t + c)) + d$ with $b > 0$ , what is $b$ ?

$\frac{\pi}{6}$
$12$
$24\pi$
$\frac{\pi}{12}$

Show the answer

$b$ $= \frac{2\pi}{\text{period}}$ $= \frac{2\pi}{12}$ $= \frac{\pi}{6}$ .

Correct: $\frac{\pi}{6}$ :
$12$ :
$24\pi$ :
$\frac{\pi}{12}$ :

2. A student runs a regression on a graphing calculator. What does the regression do?

Finds the constants of the chosen model type that fit the data most closely
Finds the exact formula the data came from
Draws the line through the first and last data points
Rounds each data value to three decimal places

Show the answer

A regression chooses the constants of a model type you pick (linear, quadratic, exponential and so on) so the model is as close as possible to the data.

Correct: Finds the constants of the chosen model type that fit the data most closely:
Finds the exact formula the data came from:
Draws the line through the first and last data points:
Rounds each data value to three decimal places:

Part 4 · See it

See it first

Left: a Ferris wheel on a support above the ground, turning counterclockwise, with a seat S at the lowest point of the wheel, which is above the ground. Right: an unscaled graph of h(t), the height of the seat above the ground, for one turn. The graph starts at its lowest point F, rises through G to its highest point J, falls through K and ends at the lowest point M. G and K are at the height of the center of the wheel. The axes have no scale. — A seat on a Ferris wheel and its height over one turn. The five labeled points (lowest, midline, highest, midline, lowest) are where a model comes from. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

The context gives the greatest and least valuesthe midline $d = \frac{\max + \min}{2}$ and the amplitude $|a| = \frac{\max - \min}{2}$
The context gives the time for one cycle, or the frequencythe period ( $\frac{1}{\text{frequency}}$ if needed), then $b = \frac{2\pi}{\text{period}}$
The context gives one landmark: when the quantity is highest, lowest, or at its midline and risingchoose sine or cosine and the sign of $a$ to match that landmark, then set $c$ so the landmark lands at the right time
The model is used to predictit is trustworthy only on the domain the context supports; with data, a sinusoidal regression on a calculator builds the same kind of model

Part 6 · Key ideas

Key ideas

Five labeled points on one cycle are: a highest or lowest point, a midline crossing, the opposite extreme, the other midline crossing and the first extreme again, each a quarter period apart.
Easy landmarks: highest at $t_0$ gives $a\cos(b(t - t_0)) + d$ with $a > 0$ ; lowest at $t_0$ gives the same with $a < 0$ ; midline and rising at $t_0$ gives $a\sin(b(t - t_0)) + d$ with $a > 0$ .
Describe change on an interval in two separate statements: whether the quantity is increasing or decreasing, and whether its rate of change is increasing or decreasing (concave up or concave down).
A model fits the facts it was built from. Past the data, or if the real cycle drifts, predictions get worse.

Part 7 · Misconception

A common mistake

The wrong idea: If a tide goes from high to low in 6 hours, the period is 6 hours.

What actually happens: High to low is half a cycle. The water must come back up to high tide to finish a cycle, so the period is 12 hours and $b = \frac{2\pi}{12} = \frac{\pi}{6}$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Temperature in a greenhouse

The temperature in a greenhouse,

T(t)

degrees Celsius,

t

hours after midnight, is modeled by a sinusoidal function. The least temperature,

12^\circ

C, occurs at 4 a.m. (

t = 4

), and the greatest,

28^\circ

C, at 4 p.m. (

t = 16

). The unscaled graph shows

T

for

4 \le t \le 28

P

and

U

are lowest points,

R

is a highest point, and

Q

and

S

are on the midline.

An unscaled graph of T(t), the temperature in a greenhouse, in degrees Celsius, t hours after midnight, for t from 4 to 28. The graph starts at a lowest point P, rises through Q to a highest point R, falls through S and ends at the next lowest point U. Q and S are halfway between the lowest and highest temperatures. The axes have no scale. — LevlPrep original diagram.

1. What are the coordinates of $R$ ?

$(16, 28)$
$(16, 20)$
$(12, 28)$
$(28, 16)$

Show the answer

$R$ is the highest point: $28^\circ$ C at $t = 16$ .

Correct: $(16, 28)$ : Time first, then temperature: 4 p.m. is $t = 16$ .
$(16, 20)$ : 20 is the midline temperature; $R$ is the highest point.
$(12, 28)$ : The greatest temperature is at 4 p.m., which is $t = 16$ , not noon.
$(28, 16)$ : The coordinates are reversed: the input $t$ comes first.

2. What are the coordinates of $S$ ?

$(22, 20)$
$(20, 22)$
$(22, 12)$
$(19, 20)$

Show the answer

The period is 24 hours (lowest at 4 and again at 28), so landmarks are 6 hours apart. $S$ is 6 hours after $R$ , on the midline $\frac{28 + 12}{2} = 20$ : $(22, 20)$ .

Correct: $(22, 20)$ : A quarter period after the highest point, at the midline height.
$(20, 22)$ : The coordinates are reversed.
$(22, 12)$ : 12 is the least temperature; $S$ is on the midline.
$(19, 20)$ : This uses 3 hours as the quarter period; the period is 24 hours, so a quarter is 6.

3. Write a formula for $T(t)$ .

Type an expression for $T(t) =$ .

Show the answer

$d = \frac{28 + 12}{2} = 20$ , $|a| = \frac{28 - 12}{2} = 8$ , period 24 so $b = \frac{2\pi}{24} = \frac{\pi}{12}$ . Lowest at $t = 4$ : a cosine with $a = -8$ shifted right 4. $T(t)$ $= -8\cos\left(\frac{\pi}{12}(t - 4)\right) + 20$ ; any equivalent form is right.

Answer: $T(t) =$ $-8\,\cos\left(\frac{\pi}{12} (t - 4)\right) + 20$

4. Which statement describes $T$ on the interval $(16, 22)$ ?

$T$ is decreasing, and the rate of change of $T$ is decreasing
$T$ is decreasing, and the rate of change of $T$ is increasing
$T$ is increasing, and the rate of change of $T$ is decreasing
$T$ is decreasing, and the rate of change of $T$ is constant

Show the answer

From $R$ to $S$ the temperature falls from its peak toward the midline, so $T$ is decreasing. The graph is above the midline, concave down, so the rate of change of $T$ is decreasing: it starts at 0 at the peak and becomes more and more negative.

Correct: $T$ is decreasing, and the rate of change of $T$ is decreasing: Falling, and concave down above the midline.
$T$ is decreasing, and the rate of change of $T$ is increasing: The rate of change is increasing only where the graph is concave up, below the midline, as on $(22, 28)$ .
$T$ is increasing, and the rate of change of $T$ is decreasing: After the highest point at $t = 16$ the temperature falls.
$T$ is decreasing, and the rate of change of $T$ is constant: A sinusoid's rate of change is never constant on an interval; here it keeps decreasing.

5. What is the average rate of change of $T$ on $[10, 16]$ , in degrees Celsius per hour?

No calculator. Type the exact value (a fraction, π, e or a root is fine), in degrees Celsius per hour.

Show the answer

$Q = (10, 20)$ and $R = (16, 28)$ : $\frac{T(16) - T(10)}{16 - 10}$ $= \frac{28 - 20}{6}$ $= \frac{4}{3}$ degrees Celsius per hour, the slope of the secant line from $Q$ to $R$ .

Answer: $\frac{4}{3}$ ≈ 1.333 degrees Celsius per hour

6. A weight on a spring bobs up and down with a frequency of 4 cycles per second. Its height is modeled by $h(t) = a\sin(bt) + d$ , $t$ in seconds, with $b > 0$ . What is $b$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Period $= \frac{1}{4}$ second, so $b = \frac{2\pi}{1/4} = 8\pi$ .

Answer: $8\,\pi$ ≈ 25.133

7. The height of a swing seat is $h(t) = 5\sin\left(\frac{\pi}{2}t\right) + 7$ feet, $t$ in seconds. Which describes $h$ on the interval $(1, 2)$ ?

$h$ is decreasing, and the rate of change of $h$ is decreasing
$h$ is decreasing, and the rate of change of $h$ is increasing
$h$ is increasing, and the rate of change of $h$ is decreasing
$h$ is increasing, and the rate of change of $h$ is increasing

Show the answer

The period is $\frac{2\pi}{\pi/2} = 4$ . The highest point is at $t = 1$ ( $h = 12$ ) and the midline crossing at $t = 2$ ( $h = 7$ ). On $(1, 2)$ the height falls while the graph is above the midline, concave down, so the rate of change of $h$ is decreasing.

Correct: $h$ is decreasing, and the rate of change of $h$ is decreasing: Falling from the top toward the midline, concave down.
$h$ is decreasing, and the rate of change of $h$ is increasing: The rate of change increases only below the midline, on $(2, 4)$ here.
$h$ is increasing, and the rate of change of $h$ is decreasing: That describes $(0, 1)$ , before the highest point.
$h$ is increasing, and the rate of change of $h$ is increasing: That describes $(3, 4)$ , rising from the lowest point.

Part 9 · Summary

Summary

To model a periodic context, get the midline and amplitude from the greatest and least values, the period (and

b = \frac{2\pi}{\text{period}}

) from the cycle length or frequency, and the shift from one landmark time. With data, sinusoidal regression does the same job. Use the model on the domain the context supports, and describe change on an interval as two statements: the direction of the quantity and the direction of its rate of change.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic