MCR3U Sinusoidal Functions Practice Test (with Solutions)

Free Trigonometric Functions Practice Test

6 questions (13 parts), exactly as a free account gets them in the first Trigonometric Functions test on Martian Lab. Try each one on paper first, then open the answer.

1. Complete the following table.

   Function	Period	Amplitude	Phase shift	Vertical displacement	Range
   $y = 3\cos(3x + 180^\circ) - 1$
   $y = \sin\frac{3}{4}(2x + 180^\circ) - 1$
   $y = \frac{1}{3}\cos\!\left(4x - \frac{90^\circ}{2}\right) + 3$
   $y = \underline{\quad} \cos \underline{\quad\quad}$	$90^\circ$	$\dfrac{1}{2}$	$-60^\circ$	$-3$
   $y = \underline{\quad} \sin \underline{\quad\quad}$	$720^\circ$	$3$	$60^\circ$	N/A

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   Function	Period	Amplitude	Phase shift	Vertical displacement	Range
   $y = 3\cos(3x + 180^\circ) - 1$	$120^\circ$	$3$	$-60^\circ$	$-1$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -4 \leq y \leq 2 \end{array}\right.$
   $y = \sin\frac{3}{4}(2x + 180^\circ) - 1$	$240^\circ$	$1$	$-90^\circ$	$-1$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -2 \leq y \leq 0 \end{array}\right.$
   $y = \frac{1}{3}\cos\!\left(4x - \frac{90^\circ}{2}\right) + 3$	$90^\circ$	$\dfrac{1}{3}$	$+11.25^\circ$	$+3$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ \frac{8}{3} \leq y \leq \frac{10}{3} \end{array}\right.$
   $y = \frac{1}{2}\cos\left[2(x+60)\right]^\circ - 3$	$90^\circ$	$\dfrac{1}{2}$	$-60^\circ$	$-3$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -\frac{7}{2} \leq y \leq -\frac{5}{2} \end{array}\right.$
   $y = 3\sin\!\left[\frac{1}{2}(x-60)\right]^\circ$	$720^\circ$	$3$	$60^\circ$	N/A	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -3 \leq y \leq 3 \end{array}\right.$

2. How many x intercepts will the graph of the equation $y = -2\sin\theta - 1$ from $-360^\circ \leq \theta \leq 360^\circ$?

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   $0 = -2\sin\theta - 1$ $1 = -2\sin\theta$ $-\tfrac{1}{2} = \sin\theta$ $\therefore\ -360^\circ \leq \theta \leq 360^\circ$ $\therefore\ \theta_1 = -30^\circ \; \theta_2 = 330^\circ$ $\theta_3 = 210^\circ \; \theta_4 = -150^\circ$ There are 4 x-intercepts.
3. A Ferris wheel at Charlotte's Amusement Park has a diameter of $18\,\mathrm{m}$ and travels at a rate of 6 revolutions every 2 min. At the bottom of the ride, the passenger is $2\,\mathrm{m}$ above the ground.

   1. Sketch the graph of the rider's height above the ground, in meters, verses the time, in seconds, for 2 minutes. Assume the rider starts at the highest position at $t = 0$. Be sure to show the 5 major points of each period, the central axis, and the appropriate labels.
      Show solution

      A correct sketch has amplitude $9$, midline $h=11$, period $20\,\mathrm{s}$, and starts at the maximum point. Major points for one period are $(0,20)$, $(5,11)$, $(10,2)$, $(15,11)$, $(20,20)$. This pattern repeats every $20\,\mathrm{s}$ for $0 \leq t \leq 120$.
   2. Write the equation of the height that represents the height $h$ above the ground at time $t$ as a cosine function.
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      $h(t) = 9\cos 18t + 11$
   3. How high above the ground will the rider be at $t = 78\,\mathrm{s}$?
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      $h(78) = 9\cos 18 \cdot 78 + 11$ $h(78) = 18.281152\,\mathrm{m}$ $h(78) \approx 18.28\,\mathrm{m}$ $\therefore$ the rider will be approximately $18.28\,\mathrm{m}$ above the ground at $t=78\,\mathrm{s}$.
4. Sketch $y = -2\sin\!\left(\dfrac{1}{3}x - 180^\circ\right) - 1$ for $-360^\circ \leq x \leq 360^\circ$

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   $y = -2\sin\!\left[\tfrac{1}{3}(x - 540)^\circ\right] - 1$ period $= 1080^\circ$ $(x,y) \rightarrow (3x + 540^\circ,\ -2y - 1)$
5. Sketch $f(x) = \tan x$ for $-360^\circ \leq x \leq 360^\circ$ and state the domain, range, period, and vertical asymptotes.

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   Domain: $\left\{x \mid x \neq \pm 90^\circ, \pm 270^\circ,\ -360^\circ \leq x \leq 360^\circ,\ x \in \mathbb{R}\right\}$ Range: $\left\{y \mid y \in \mathbb{R}\right\}$ Period: $180^\circ$ V.A. $= -270^\circ$ V.A. $= -90^\circ$ V.A. $= 90^\circ$ V.A. $= 270^\circ$
6. Given the following sinusoidal function, find:

   

   1. Amplitude
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      $\dfrac{1}{2}$
   2. Period
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      $360^\circ$
   3. Eqn of Axis of Symmetry
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      $y = 1$
   4. k value
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      $1$
   5. Equation for the sine curve
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      $y = \dfrac{1}{2}\sin(x - 90^\circ) + 1$
   6. Equation for the consine curve
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      $y = -\dfrac{1}{2}\cos x + 1$

This Unit on Martian Lab

The lessons go from graphing trigonometric functions to writing equations of sinusoidal functions and applying them.

1. Graphing Trignometric Functions
2. Equations of Sinusoidal Functions
3. Applications of Sinusoidal Functions

Then 4 full Trigonometric Functions practice tests to check yourself.

Trigonometric Functions FAQ

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