Graphs of Trigonometric Functions: Algebra 2 Math Test – PDF Download

Question 1: Features of a sinusoid (2 points)

Consider \(y=-2\sin(3(x-\frac{\pi}{6}))+4\).

1. State the amplitude and the midline. (1 point)
2. State the period and the phase shift. (1 point)

Question 2: Equation from a graph (2 points)

The graph shows a sine wave.

1. State the midline and the amplitude. (1 point)
2. Write an equation of the form \(y=A\sin(Bx)+D\) for the graph. (1 point)

Question 3: Evaluate exactly (2 points)

Let \(f(x)=4\cos x-2\). Find the exact value of each.

1. \(f(\frac{\pi}{3})\) (1 point)
2. \(f(\pi)\) (1 point)

Question 4: Write a cosine model (2 points)

A cosine function has a maximum value of \(9\), a minimum value of \(-1\) and a period of \(6\). Write an equation of the form \(y=A\cos(Bx)+D\) with \(A > 0\) and \(B > 0\). Show how you found each constant.

Question 5: Phase shift from a graph (2 points)

The graph shows \(y=A\sin(x-C)\) with \(A > 0\) and \(0 < C < \pi\).

Find \(A\) and \(C\), and write the equation.

Question 6: Compare two sine waves (2 points)

Compare \(f(x)=\sin x\) and \(g(x)=\sin 2x\) on the interval \([0,4\pi]\).

1. State the period of each function. (1 point)
2. How many complete cycles does each graph make on the interval? (1 point)

Question 7: Ferris wheel height (3 points)

Priya’s Ferris wheel has height \(h(t)=30-22\cos(\frac{\pi t}{18})\) feet, where \(t\) is in seconds.

1. Find the lowest and highest heights. (1 point)
2. How long does one full turn take? (1 point)
3. Find the height at \(t=6\) seconds. (1 point)

Question 8: Harbor depth model (3 points)

The graph shows the depth of water at a harbor on one day, where \(t\) is the number of hours after midnight. The depth follows \(D(t)=A\cos(B(t-C))+K\) with \(A > 0\), \(B > 0\) and \(C > 0\).

1. Find the amplitude, midline and period. (1 point)
2. Write the function \(D(t)\). (1 point)
3. Find the depth at \(t=5\). (1 point)

Question 9: Temperature model (2 points)

The temperature in Diego’s town is \(T(t)=60+12\sin(\frac{\pi(t-8)}{12})\) degrees Fahrenheit, where \(t\) is the number of hours after midnight.

1. Find the maximum temperature and the time it first occurs. (1 point)
2. Find the temperature at \(t=11\) to the nearest tenth. (1 point)

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