Precalculus Graphs of Trigonometric Functions Test Solutions and Scoring (PDF)

Precalculus math test solutions: Graphs of Trigonometric Functions – Maths PDF

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This precalculus trig graphs test answers for Precalculus covers graphs of trigonometric functions (Common Core F-TF.B.5, F-IF.C.7e, F-TF.B.7), with a free PDF to print.

Worked solutions and scoring notes for the 9-question trig graphs test. Use the scoring line under each solution to award partial credit.

Precalculus · Common Core State Standards: F-TF.B.5, F-IF.C.7e, F-TF.B.7

Precalculus trig graphs test answers: worked solutions

Question 1 – Read the equation [2 pts]

Amplitude \(3\). \(b=\frac{\pi}{2}\), so the period is \(\frac{2\pi}{\pi/2}=4\). Midline \(y=4\).

\(3\); \(4\); \(y=4\)

Scoring: 1 point for amplitude and midline, 1 point for the period.

Question 2 – Phase shift [2 pts]

\(\cos\left(2\left(x-\frac{\pi}{4}\right)\right)\). Period \(\frac{2\pi}{2}=\pi\). Shift \(\frac{\pi}{4}\) right.

\(\cos(2(x-\frac{\pi}{4}))\); period \(\pi\); shift \(\frac{\pi}{4}\) right

Scoring: 1 point for factoring, 1 point for period and shift.

Question 3 – Evaluate [2 pts]

\(4\cos0=4\). \(4\cos\frac{\pi}{2}=0\). \(4\cos\pi=-4\).

\(4,\ 0,\ -4\)

Scoring: 1 point for two correct values, 2 points for all three.

Question 4 – Tangent [2 pts]

Period \(\frac{\pi}{3}\). Asymptote when \(3x=\frac{\pi}{2}\).

\(\frac{\pi}{3}\); \(x=\frac{\pi}{6}\)

Scoring: 1 point each.

Question 5 – Read a trig graph [3 pts]

Max 4, min -2, period pi.

(a) Amplitude \(\frac{4-(-2)}{2}=3\); period \(\pi\). (b) Midline \(y=1\); range \([-2,4]\). (c) \(b=\frac{2\pi}{\pi}=2\).

\(3,\ \pi\); \(y=1\), \([-2,4]\); \(y=3\cos2x+1\)

Scoring: 1 point for each part.

Question 6 – Write an equation [2 pts]

\(b=\frac{2\pi}{6\pi}=\frac13\).

\(y=4\sin\frac{x}{3}\)

Scoring: 1 point for the amplitude, 1 point for b.

Question 7 – Range [2 pts]

\(-2\sin x\) ranges over \([-2,2]\). Add 3.

\([1,5]\)

Scoring: 1 point for the amplitude step, 1 point for the range.

Question 8 – Tide model [3 pts]

Depth is 7 ft at t = 2 and t = 10.

(a) \(d(3)=6+2\cos\frac{\pi}{2}=6\). (b) \(\frac{2\pi}{\pi/6}=12\) h. (c) \(2\cos\frac{\pi t}{6}=1\), so \(\cos\frac{\pi t}{6}=\frac12\), \(\frac{\pi t}{6}=\frac{\pi}{3}\) or \(\frac{5\pi}{3}\), giving \(t=2\) or \(t=10\).

\(6\) ft; \(12\) h; \(t=2\) and \(t=10\)

Scoring: 1 point for each part.

Question 9 – Ferris wheel [2 pts]

Greatest height \(25+20=45\) ft. \(h(10)=25-20\cos\frac{\pi}{2}=25\) ft.

\(45\) ft; \(25\) ft

Scoring: 1 point each.

Total score

Total

20 points

Key facts behind this precalculus trig graphs test answers

Key facts

Amplitude = |a| = (max – min)/2; midline y = k = (max + min)/2.
Period of sine and cosine = 2 pi / |b|; period of tangent = pi / |b|.
Phase shift h moves the graph right when h is positive.
Tangent has vertical asymptotes at x = pi/2 + k pi.
Use cosine for a model that starts at a maximum and sine for one that starts at the midline.

How to use this precalculus trig graphs test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Precalculus chapters: Precalculus math.

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