Key takeaway: This precalculus trig graphs lesson shows how to read and draw sine, cosine and tangent graphs. A sinusoid \(y=A\sin(B(x-C))+D\) has amplitude \(|A|\), period \(\frac{2\pi}{|B|}\), phase shift \(C\) and midline \(y=D\). Tangent repeats every \(\pi\) and has vertical asymptotes instead of peaks. The notes follow standard F-TF.B.5, and they end with a method for writing a model from data.
A sinusoid is a sine or cosine wave written as y = A sin(B(x − C)) + D. Its amplitude is |A|, its period is 2π/|B|, its phase shift is C, and its midline is y = D. The tangent graph repeats every π and has vertical asymptotes instead of a maximum or minimum.
Precalculus Trig Graphs Lesson: key facts
- Amplitude is half the distance between the maximum and the minimum.
- The period of sine and cosine is 2π divided by |B|; the period of tangent is π divided by |B|.
- A positive C shifts the graph right; a positive D shifts it up.
- Tangent has vertical asymptotes at x = π/2 + kπ and no amplitude.
- A periodic data set with a known maximum and minimum fits a model with A = (M − m)/2 and D = (M + m)/2.
What do the sine and cosine graphs look like?
The graphs of \(y=\sin x\) and \(y=\cos x\) are smooth waves that repeat every \(2\pi\) radians. Both stay between \(-1\) and \(1\). The cosine wave is the sine wave moved left by \(\frac{\pi}{2}\).
How do amplitude, period, and shifts change the graph?
\(y=A\sin(B(x-C))+D\) or \(y=A\cos(B(x-C))+D\)
- Amplitude: \(|A|\), the distance from the midline to a peak.
- Period: \(\frac{2\pi}{|B|}\), the length of one full cycle.
- Phase shift: \(C\), right if positive, left if negative.
- Midline: \(y=D\), the vertical shift.
How do you graph a sinusoid step by step?
- Read \(A\), \(B\), \(C\), and \(D\) from the equation.
- Draw the midline \(y=D\) and the lines \(y=D\pm|A|\).
- Find the period \(\frac{2\pi}{|B|}\) and divide it into four equal steps.
- Plot the five key points, then sketch the wave.
Example. Graph \(y=-2\cos(\frac{x}{2})+1\). Here \(|A|=2\), the period is \(\frac{2\pi}{1/2}=4\pi\), and the midline is \(y=1\). The negative sign flips the cosine, so the wave starts at its minimum \(-1\) and has key points every \(\pi\).
What does the tangent graph look like?
The graph of \(y=\tan x\) has period \(\pi\), crosses the x-axis at multiples of \(\pi\), and has vertical asymptotes at \(x=\frac{\pi}{2}+k\pi\). For \(y=A\tan(Bx)\) the period is \(\frac{\pi}{|B|}\). It has no amplitude because it is unbounded.
How do you write a sinusoidal model from data?
Use the maximum \(M\) and minimum \(m\): \(A=\frac{M-m}{2}\) and \(D=\frac{M+m}{2}\). The time from a maximum to the next minimum is half a period, so \(B=\frac{2\pi}{\text{period}}\). A tide rises to 9 ft at hour 2 and drops to 1 ft at hour 8. Then \(A=4\), \(D=5\), the period is 12, and \(h=4\cos(\frac{\pi}{6}(t-2))+5\).
Graphs of Trigonometric Functions: frequently asked questions
What is the difference between amplitude and period?
Amplitude measures how tall the wave is: the distance from the midline to a peak. Period measures how long one full cycle takes along the x-axis.
How do you find the phase shift of y = sin(2x − π)?
Factor the inside as 2(x − π/2). The phase shift is π/2 to the right. Always factor out B before reading the shift.
Why does tangent have asymptotes?
Tangent equals sine divided by cosine. At x = π/2 + kπ the cosine is zero, so the quotient is undefined and the graph climbs without bound.
Can a sine function be written as a cosine function?
Yes. Cosine is sine shifted left by π/2, so sin x = cos(x − π/2). Both can describe the same wave with different phase shifts.
How do you choose between sine and cosine for a model?
Use cosine when the data starts at a maximum or minimum, and sine when it starts on the midline. Either works if you pick the right shift.
How do amplitude and period change a sinusoid?
Take \(y=-5\sin 3x+1\). The amplitude is 5, because the sign only reflects the wave. The period is \(\frac{2\pi}{3}\), so the wave repeats three times faster than \(\sin x\). Finally, the midline \(y=1\) moves the whole graph up.
How do you find the phase shift of a cosine?
Factor out the coefficient of \(x\) first. For \(y=\cos(2x+\pi)\), write \(\cos(2(x+\frac{\pi}{2}))\). Therefore the shift is \(\frac{\pi}{2}\) to the left. Many students read the shift as \(\pi\), and that mistake skips the factoring step.
What does the tangent graph look like in a precalculus trig graphs lesson?
The curve rises through the origin and climbs toward asymptotes at \(x=\pm\frac{\pi}{2}\). Because it has no maximum, tangent has no amplitude. For \(y=3\tan 2x\), the period shrinks to \(\frac{\pi}{2}\), so the closest asymptotes sit at \(\pm\frac{\pi}{4}\).
How do you write a model from a maximum and a minimum?
Suppose a wave peaks at 10 and dips to 4. Then the amplitude is 3 and the midline is \(y=7\). Next, choose cosine when the graph starts at a peak. For a 12-hour period, \(B=\frac{\pi}{6}\), which gives \(h=3\cos(\frac{\pi t}{6})+7\).
What is the best way to rehearse trig graphs?
Sketch five key points for three different sinusoids before you open any problem set. Then change one parameter at a time and predict the effect. After that, graph one tangent with a shift, and compare each sketch with a graphing tool.
See every chapter on the Precalculus math page.
