Exercises for Question Type 15: Formulating sinusoidal models given descriptions

Question 1

Skill question

In radians, write $y = A sin (B (x - C)) + D$ for a sinusoid with amplitude $5$ , period $\frac{2 π}{3}$ , shifted $\frac{π}{4}$ to the left, and midline $y = 1$ .

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Question 2

Skill question

Find a model $y = A sin (B x) + D$ with amplitude $3$ , period $18 0^{\circ}$ , and midline $y = 2$ , assuming no horizontal shift.

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Question 3

Skill question

A daily temperature oscillation has amplitude $1 0^{\circ} C$ , average $2 0^{\circ} C$ , and peaks at 3 pm. Model the temperature $T (t)$ in the form $T = 10 sin (B (t - C)) + 20$ , where $t$ is hours after midnight.

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Question 4

Skill question

A mass on a spring undergoes simple harmonic motion described by $d (t) = A sin (ω t + ϕ)$ . If the amplitude is $3 cm$ , the period is $4 s$ , and the mass passes upward through equilibrium at $t = 1 s$ , find $d (t)$ .

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Question 5

Skill question

The height of the tide in a harbour, $h$ metres, can be modelled by the function $h (t) = 2 sin (B (t - C)) + 5$ where $t$ is the time in hours after midnight. The tide has an amplitude of $2$ metres, a mean level of $5$ metres, and a period of $12$ hours. A high tide occurs at $t = 6$ .

Find the expression for $h (t)$ by determining the values of $B$ and $C$ .

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Question 6

Skill question

A Ferris wheel has radius $20$ m, with its centre $22$ m above the ground, and completes one revolution per minute. A passenger starts at the lowest point at $t = 0$ . Write the height $H (t)$ in metres at time $t$ minutes in the form $H (t) = 20 sin (B t + ϕ) + 22$ .

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Question 7

Skill question

Model the average daylight hours $D (d)$ over a year with amplitude $2$ , mean $12$ , maximum on day $172$ , and period $365$ days, in the form $D = 2 sin (\frac{2 π}{365} (d - C)) + 12$ .

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Question 8

Skill question

Write a model $y = A sin (B x + C) + D$ for a sinusoid with amplitude $2$ , period $6 0^{\circ}$ , midline $y = 0$ , and shifted $1 5^{\circ}$ to the left.

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Question 9

Skill question

Determine the sinusoidal model $y = A sin (B (x - C)) + D$ with amplitude $4$ , period $12 0^{\circ}$ , midline $y = - 1$ , and phase shift $3 0^{\circ}$ to the right.

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Question 10

Skill question

A sinusoid has a maximum value of $8$ at $x = 1 0^{\circ}$ and a minimum value of $2$ at $x = 10 0^{\circ}$ , repeating every $18 0^{\circ}$ . Find its equation in the form $y = A sin (B (x - C)) + D$ .

Find the equation of the sinusoid in the form $y = A sin (B (x - C)) + D$ .

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Question 11

Skill question

Determine the equation $y = A sin (B (x - C)) + D$ for a sinusoid with amplitude $6$ , period $10 0^{\circ}$ , midline $y = - 2$ , phase shift $2 0^{\circ}$ right, and reflected across its midline.

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Question 12

Skill question

A sinusoidal model of the form $y = A sin (B x) + D$ is defined by its amplitude, period, and principal axis.

Formulate a sinusoidal model of the form $y = A sin (B x) + D$ that has amplitude $5$ , period $9 0^{\circ}$ , and principal axis $y = 11$ , with no phase shift.

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IB Mathematics Applications & Interpretation Question Type 15: Formulating Sinusoidal Models Given Descriptions Exercises