Graphs of Trigonometric Functions: math test, Grade 12 – download the PDF

1 Analyzing a sinusoid / 4 pts

Let \(y=-2\sin(3x-\pi)+1\).

1. Rewrite the function in the form \(A\sin(B(x-C))+D\) and give the amplitude and the period.
2. Give the phase shift with its direction.
3. Give the midline, the maximum and the minimum.
4. Compute \(y\) at \(x=\dfrac{\pi}{3}\) and at \(x=\dfrac{\pi}{2}\).

2 Writing an equation / 4 pts

A cosine function has a maximum value of 12 at \(x=0\) and its next minimum value of 2 at \(x=3\).

1. Find its amplitude, midline and period.
2. Write its equation.
3. Compute \(y(1)\) and find all \(x\) in \([0,6]\) with \(y=9.5\).

3 Tangent and cotangent / 3 pts

1. For \(y=\tan(3x)\): give the period, the asymptotes nearest the origin, and \(y\) at \(x=\dfrac{\pi}{12}\).
2. For \(y=\cot x\): give the asymptotes in \([0,\pi]\), the zero in \((0,\pi)\), and the values at \(x=\dfrac{\pi}{4}\) and \(x=\dfrac{3\pi}{4}\).

4 Cosecant / 3 pts

Let \(y=3\csc x\).

1. Give the asymptotes and the range.
2. Compute \(y\) at \(x=\dfrac{\pi}{6}\) and at \(x=\dfrac{7\pi}{6}\).
3. Identify the local extrema on \((0,2\pi)\).

5 Inverse trigonometric values / 3 pts

Evaluate exactly.

1. \(\arcsin\dfrac{\sqrt3}{2}\)
2. \(\arccos\dfrac{\sqrt2}{2}\)
3. \(\arctan(-\sqrt3)\)
4. \(\cos\left(\arcsin\dfrac{5}{13}\right)\)
5. \(\arcsin\left(\sin\dfrac{7\pi}{6}\right)\)

6 Hours of daylight / 3 pts

Calculator allowed. In a northern city, the hours of daylight on day \(t\) of the year (\(t=1\) is January 1) are modeled by \(D(t)=12+3.5\sin\left(\dfrac{2\pi(t-80)}{365}\right)\).

1. Give the maximum and the minimum number of daylight hours, and the period.
2. Compute \(D(172)\) to the nearest hundredth.
3. For about how many days of the year are there at least 14 hours of daylight?