Sinusoidal Functions Practice

These questions follow Sinusoidal Functions — reading equations, building them from properties, and running models like the one your group built for The Ferris Wheel. Angles are in degrees throughout.

Reading equations

1. For $y=4\sin x$: state the amplitude, period, maximum, and minimum.
2. For $y=3\sin (2x)-1$: state the amplitude, period, equation of the axis, maximum, minimum, and range.
3. For $y=2\cos (x-60°)+5$: state the amplitude, phase shift, equation of the axis, and range.

Answer 1

Amplitude 4; period $360°$ (no $k$, no change); maximum 4 and minimum $-4$, since the axis is still $y=0$.

Answer 2

Amplitude 3; period $\frac{360°}{2}=180°$; axis $y=-1$. Maximum $-1+3=2$, minimum $-1-3=-4$; range $-4\le y\le 2$. Every answer hangs off two numbers: the axis and the amplitude.

Answer 3

Amplitude 2; phase shift $60°$ right; axis $y=5$; range $3\le y\le 7$. The period is untouched at $360°$ — no $k$ inside the bracket.

Building equations

4. A sinusoidal function has amplitude 2, period $180°$, and a maximum point at $(0,3)$. Represent it with an equation in two different ways.
5. (a) What is the period of $y=\sin (3x)$? (b) What value of $k$ gives a sinusoid a period of $720°$?

Answer 4

Period $180°$ forces $k=2$; a maximum of 3 with amplitude 2 puts the axis at $y=1$. A cosine starts at its peak, so $y=2\cos (2x)+1$ works immediately. For a sine version, shift so the peak lands at $x=0$: $y=2\sin (2(x+45°))+1$ — check: at $x=0$ the argument is $90°$, where sine peaks. ✓ Same graph, two names.

Answer 5

(a) $\frac{360°}{3}=120°$. (b) Solve $\frac{360°}{k}=720°$: $k=\frac{1}{2}$. A $k$ below 1 stretches the wave — inside letters act backwards, here as everywhere.

Models in motion

6. A Ferris wheel rider’s height is $h(t)=25\sin (3(t-30))+27$, with $h$ in metres and $t$ in seconds. Find the maximum and minimum heights, the height at $t=30$, and the time for one full revolution.
7. A tide follows $h(t)=5\sin (30(t+3))$, with $h$ in metres and $t$ in hours after midnight. Find the period, verify that high tide is at midnight, and find the first low tide after midnight.
8. For the wheel in question 6: what changes in the equation if the wheel turns twice as fast, and what if the boarding platform is raised by 1 m?

Answer 6

Axis 27, amplitude 25: maximum $52$ m, minimum $2$ m. At $t=30$ the sine’s argument is $0°$, so $h(30)=27$ m — the rider crosses the axis. One revolution is the period: $\frac{360°}{3}=120$ seconds.

Answer 7

Period: $\frac{360°}{30}=12$ hours. High tide: sine peaks when its argument is $90°$, and $30(t+3)=90$ gives $t=0$ — midnight. ✓ Low tide: argument $270°$ gives $30(t+3)=270$, so $t=6$ — 6 a.m., half a cycle later, as it must be.

Answer 8

Twice as fast doubles $k$ from 3 to 6, halving the period to 60 s — amplitude and axis untouched, since the wheel itself did not change size. Raising the platform 1 m lifts everything: $c$ goes from 27 to 28. Each physical change edits exactly one letter; that one-to-one mapping is the whole point of the form.