Graphs of Trigonometric Functions: Precalculus Math Worksheet – PDF Download

Precalculus Trig Graphs Worksheet: problems 1–5

Problem 1: Amplitude of a wave

State the amplitude of each function.

1. \(y=4\sin x\)
2. \(y=-3\cos x\)
3. \(y=\frac{1}{2}\sin 2x\)
4. \(y=5\cos x+2\)

Problem 2: Period from B

Find the period of each function.

1. \(y=\sin 2x\)
2. \(y=\cos(\frac{x}{2})\)
3. \(y=\sin 3x\)
4. \(y=\cos(\pi x)\)

Problem 3: Midline and range

State the midline and the range of each function.

1. \(y=2\sin x+3\)
2. \(y=-4\cos x-1\)
3. \(y=\frac{1}{2}\sin x-2\)
4. \(y=3\cos 2x+0.5\)

Problem 4: Five key points

Evaluate each function at \(x=0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi\).

1. \(y=2\cos x+1\)
2. \(y=-\sin x\)

Problem 5: Read a cosine graph

The graph shows a cosine wave. State the amplitude, period, and midline, then write an equation of the form \(y=A\cos(Bx)+D\).

Precalculus Trig Graphs Worksheet: problems 6–10

Problem 6: Graph a reflected sine

Graph one full period of \(y=-2\sin x\) on \(0\le x\le 2\pi\). Label the five key points.

Problem 7: Describe transformations

Describe how the graph of each function is obtained from its parent function.

1. \(y=-\frac{1}{2}\cos x+3 \text{ from } y=\cos x\)
2. \(y=\sin(x+\frac{\pi}{2}) \text{ from } y=\sin x\)
3. \(y=\cos(\frac{x}{4}) \text{ from } y=\cos x\)

Problem 8: Phase shift

State the phase shift, with direction, of each function.

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

Problem 9: Read a tangent graph

The graph shows a function of the form \(y=A\tan(Bx)\). State its period and its asymptotes, then write its equation.

Problem 10: Tangent period and asymptotes

For each function, find the period and the two asymptotes closest to the y-axis.

1. \(y=\tan 3x\)
2. \(y=\tan(\frac{x}{2})\)

Precalculus Trig Graphs Worksheet: problems 11–15

Problem 11: Write a sine from a graph

The graph shows one cycle of a sine wave. Write an equation of the form \(y=A\sin(B(x-C))+D\).

Problem 12: Evaluate a sinusoid

Let \(f(x)=3\sin 2x+1\). Find each value.

1. \(f(\frac{\pi}{12})\)
2. \(f(\frac{\pi}{4})\)
3. \(f(\frac{\pi}{2})\)
4. \(f(\frac{3\pi}{4})\)

Problem 13: Write an equation

Write a sine or cosine function that has the given features.

1. Amplitude 5, period \(\pi\), midline \(y=-2\), passes through \((0,-2)\) going up.
2. Amplitude 3, period \(8\pi\), midline \(y=1\), maximum at \(x=0\).
3. Amplitude 2, period 6, midline \(y=0\), passes through \((0,0)\) going up.

Problem 14: Identify B from a graph

The graph shows a sine wave that is reflected. State the amplitude and period, then write its equation in the form \(y=A\sin(Bx)\).

Problem 15: Shifted tangent

Consider \(y=2\tan(x-\frac{\pi}{4})+1\). Find the asymptotes between \(x=-\frac{\pi}{2}\) and \(x=\pi\), the center point, and the points where \(x=0\) and \(x=\frac{\pi}{2}\). Then sketch the graph.

Precalculus Trig Graphs Worksheet: problems 16–20

Problem 16: Domain of a tangent

State the domain of \(y=\tan 2x\).

Problem 17: Ferris wheel height

Maya rides a Ferris wheel. Her height in feet is \(h(t)=-40\cos(\frac{\pi t}{2})+50\), where \(t\) is in minutes.

1. State the amplitude, period, and midline.
2. Find her height at \(t=1\) and at \(t=2\).
3. Find her height at \(t=0.5\) to the nearest tenth of a foot.

Problem 18: Tide model

The graph shows the height \(h\) of the tide, in feet, at a harbor. High tide is 11 ft at hour 3 and low tide is 3 ft at hour 9.

1. Find \(A\), \(D\), the period, and \(B\).
2. Write \(h(t)\) as a cosine function.
3. Find the height at \(t=1\) and at \(t=6\).

Problem 19: Monthly temperature

In Denver the average temperature is coldest in January (\(30^{\circ}\text{F}\), month 1) and warmest in July (\(74^{\circ}\text{F}\), month 7). Write a cosine model \(T(m)\) and use it to predict the average temperature in April (\(m=4\)) and September (\(m=9\)).

Problem 20: Bobbing buoy

A buoy rises and falls \(1.5\) ft above and below its resting level and completes one cycle every 5 seconds. At \(t=0\) it is at rest level moving up. Write a model \(y(t)\) and find the displacement at \(t=1.25\), \(t=2.5\), and \(t=0.625\) (nearest tenth).

Precalculus Trig Graphs Worksheet: problems 21–22

Problem 21: Hours of daylight

In Seattle the length of daylight \(D(t)\) in hours is modeled by \(D(t)=4\cos(\frac{2\pi}{365}(t-172))+12\), where \(t\) is the day of the year.

1. State the amplitude, midline, and period.
2. Find \(D(80)\) (March 21) and \(D(1)\) (January 1) to the nearest tenth of an hour.

Problem 22: Lighthouse beam

A lighthouse stands 300 ft from a straight shore. Its beam makes angle \(\theta\) with the line perpendicular to the shore, so the beam hits the shore a distance \(d=300\tan\theta\) feet from the point straight across.

1. Find \(d\) for \(\theta=\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) (nearest tenth).
2. Find \(d\) when \(\theta=1.5\) radians. Explain what happens as \(\theta\) approaches \(\frac{\pi}{2}\).

How to use this precalculus trig graphs worksheet

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