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

Algebra 2 Trig Test Answer Key: questions 1–5

Question 1: Features of a sinusoid (2 points)

1. \(|A|=|-2|=\) 2; midline \(y=4\).
2. Period \(\frac{2\pi}{3}\): \(\frac{2\pi}{3}\); the factored form shows a shift right \(\frac{\pi}{6}\).

Scoring: 1 point for amplitude and midline; 1 point for period and phase shift (both values in each part needed).

Question 2: Equation from a graph (2 points)

1. Max \(2\), min \(-4\): midline \(\frac{2+(-4)}{2}=\) \(y=-1\), amplitude \(\frac{2-(-4)}{2}=\) 3.
2. One cycle spans \(0\) to \(\pi\), so \(B=\frac{2\pi}{\pi}=2\). The wave rises first from the midline at \(x=0\), so \(y=3\sin(2x)-1\).

Scoring: 1 point for midline and amplitude; 1 point for the correct equation (A, B and D all correct).

Question 3: Evaluate exactly (2 points)

1. \(4\cos\frac{\pi}{3}-2=4\cdot\frac{1}{2}-2=\) 0.
2. \(4\cos\pi-2=4(-1)-2=\) \(-6\).

Scoring: 1 point for each correct exact value, with the substitution shown.

Question 4: Write a cosine model (2 points)

\(A=\frac{9-(-1)}{2}=5\); \(D=\frac{9+(-1)}{2}=4\); \(B=\frac{2\pi}{6}=\frac{\pi}{3}\).

\(y=5\cos(\frac{\pi}{3}x)+4\)

Check: \(x=0\) gives \(9\) and \(x=3\) gives \(5\cos\pi+4=-1\).

Scoring: 1 point for \(A\) and \(D\); 1 point for \(B\) and the complete equation.

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

The maximum is \(3\), so \(A=3\). The wave crosses the midline going up at \(x=\frac{\pi}{4}\), so \(C=\frac{\pi}{4}\) (check: maximum at \(\frac{\pi}{4}+\frac{\pi}{2}=\frac{3\pi}{4}\)).

\(A=3\), \(C=\frac{\pi}{4}\), \(y=3\sin(x-\frac{\pi}{4})\)

Scoring: 1 point for \(A\); 1 point for \(C\) and the equation.

Algebra 2 Trig Test Answer Key: questions 6–9

Question 6: Compare two sine waves (2 points)

1. \(f\): \(2\pi\). \(g\): \(\frac{2\pi}{2}=\) \(\pi\).
2. Cycles \(=\frac{4\pi}{\text{period}}\): \(f\) makes 2 cycles and \(g\) makes 4 cycles.

Scoring: 1 point for both periods; 1 point for both cycle counts.

Question 7: Ferris wheel height (3 points)

1. Lowest \(30-22=\) 8 ft; highest \(30+22=\) 52 ft.
2. Period \(\frac{2\pi}{\pi/18}=\) 36 seconds.
3. \(h(6)=30-22\cos\frac{\pi}{3}=30-11=\) 19 ft.

Scoring: 1 point per part.

Question 8: Harbor depth model (3 points)

1. Max \(14\), min \(6\): amplitude 4, midline \(y=10\). Two maxima at \(t=3\) and \(t=15\): period 12 hours.
2. \(B=\frac{2\pi}{12}=\frac{\pi}{6}\). The first maximum is at \(t=3\), so \(C=3\): \(D(t)=4\cos(\frac{\pi}{6}(t-3))+10\).
3. \(D(5)=4\cos\frac{\pi}{3}+10=2+10=\) 12 ft.

Scoring: 1 point per part (part a needs all three values).

Question 9: Temperature model (2 points)

1. The maximum is \(60+12=\) 72\(^{\circ}\)F when \(\frac{\pi(t-8)}{12}=\frac{\pi}{2}\), so first at \(t=14\) (2 p.m.).
2. \(T(11)=60+12\sin\frac{\pi}{4}\approx 60+8.49=\) 68.5\(^{\circ}\)F.

Scoring: 1 point for the maximum and its time; 1 point for the temperature at \(t=11\).

Scoring summary

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