Precalculus Graphs of Trigonometric Functions Worksheet Solutions (PDF)
This precalculus trig graphs 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.
Precalculus · Common Core State Standards: F-TF.B.5, F-IF.C.7e, F-TF.B.7
Precalculus trig graphs answers: exercises 1 to 10
Exercise 1 – Amplitude and period
Amplitude \(|3|=3\). Period \(\frac{2\pi}{2}=\pi\).
Amplitude \(3\); period \(\pi\)
Exercise 2 – Reflected cosine
Amplitude \(\frac12\). Here \(b=\frac13\), so the period is \(\frac{2\pi}{1/3}=6\pi\). The negative sign reflects the graph over the \(x\)-axis.
Amplitude \(\frac12\); period \(6\pi\); reflected
Exercise 3 – Period with pi in b
\(\frac{2\pi}{\pi}=2\).
2
Exercise 4 – Phase shifts
(a) \(\frac{\pi}{3}\) to the right. (b) \(\cos(2x+\pi)=\cos\left(2\left(x+\frac{\pi}{2}\right)\right)\): shift \(\frac{\pi}{2}\) left.
(a) right \(\frac{\pi}{3}\); (b) left \(\frac{\pi}{2}\)
Exercise 5 – Midline and range
Midline \(y=-3\). Maximum \(-3+2=-1\), minimum \(-3-2=-5\).
Midline \(y=-3\); range \([-5,-1]\)
Exercise 6 – Write an equation
\(b=\frac{2\pi}{\pi}=2\).
\(y=2\sin2x+1\)
Exercise 7 – Evaluate
\(3\sin\frac{\pi}{2}=3\). \(3\sin\frac{\pi}{3}=\frac{3\sqrt3}{2}\).
\(3\) and \(\frac{3\sqrt3}{2}\)
Exercise 8 – Table of values
\(2\cos0=2\), \(2\cos\frac{\pi}{3}=1\), \(2\cos\frac{\pi}{2}=0\), \(2\cos\pi=-2\).
\(2,\ 1,\ 0,\ -2\)
Exercise 9 – Read a graph
The peak height is 3. The curve completes a cycle in \(\pi\). The first maximum occurs a quarter period in: \(\frac{\pi}{4}\).
Amplitude \(3\); period \(\pi\); \(\left(\frac{\pi}{4},3\right)\)
Exercise 10 – Tangent with a stretch
Period \(\frac{\pi}{1/2}=2\pi\). Asymptotes where \(\frac{x}{2}=\pm\frac{\pi}{2}\): \(x=\pm\pi\).
Period \(2\pi\); asymptotes \(x=-\pi,\ \pi\)
Exercises 11 and up
Exercise 11 – Tangent with compression
Period \(\frac{\pi}{2}\). The first asymptote is where \(2x=\frac{\pi}{2}\).
Period \(\frac{\pi}{2}\); \(x=\frac{\pi}{4}\)
Exercise 12 – Equation from a graph (cosine)
Midline \(k=\frac{5+(-1)}{2}=2\). Amplitude \(\frac{5-(-1)}{2}=3\). The half period is \(\pi\), so the period is \(2\pi\) and \(b=1\).
\(y=3\cos x+2\)
Exercise 13 – Equation from a graph (shifted sine)
A sine curve rises through the midline at the start of a cycle, so the phase shift is \(\frac{\pi}{4}\) to the right. Check: the maximum is a quarter period later, \(\frac{\pi}{4}+\frac{\pi}{2}=\frac{3\pi}{4}\).
\(y=2\sin\left(x-\frac{\pi}{4}\right)+1\)
Exercise 14 – Range
\(\cos x\) ranges over \([-1,1]\), so \(-4\cos x\) ranges over \([-4,4]\). Add 1.
\([-3,5]\)
Exercise 15 – Phase shift identity
\(\sin\frac{\pi}{2}=1=\cos0\). \(\sin\frac{5\pi}{6}=\frac12=\cos\frac{\pi}{3}\). \(\sin\frac{3\pi}{2}=-1=\cos\pi\).
Shifting sine left pi/2 gives cosine
Exercise 16 – Alternating current
Amplitude \(170\). Period \(\frac{2\pi}{120\pi}=\frac{1}{60}\) s. Frequency \(=60\) cycles per second.
\(170\) V; \(\frac{1}{60}\) s; \(60\) cycles per second
Exercise 17 – Tide
Maximum \(5+2=7\), minimum \(5-2=3\). Period \(\frac{2\pi}{\pi/6}=12\) h. \(d(2)=5+2\cos\frac{\pi}{3}=5+1=6\).
\(7\) ft; \(3\) ft; \(12\) h; \(6\) ft
Exercise 18 – Daylight hours
Amplitude \(2.5\). The maximum is \(12+2.5=14.5\) hours. \(D(80)=12+2.5\sin0=12\).
\(2.5\); \(14.5\) h; \(12\) h
Exercise 19 – Ferris wheel
Lowest \(30-25=5\) ft, highest \(55\) ft. Period \(\frac{2\pi}{\pi/15}=30\) s. \(h(10)=30-25\cos\frac{2\pi}{3}=30+12.5=42.5\).
\(5\) ft; \(55\) ft; \(30\) s; \(42.5\) ft
Exercise 20 – A model from data
Midline \(3\), amplitude \(4\). Half period \(4\pi\), so period \(8\pi\) and \(b=\frac{2\pi}{8\pi}=\frac14\). Check: at \(x=4\pi\), \(4\cos\pi+3=-1\).
\(y=4\cos\frac{x}{4}+3\)
Key facts behind this precalculus trig graphs answers
- 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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