Precalculus Graphs of Trigonometric Functions Lesson (Free PDF)

Precalculus math lesson: Graphs of Trigonometric Functions – Maths PDF

Cover of the book Precalculus Math Lessons (PDF)

Download as a PDFThe Precalculus math lessons bookPrintable PDFSee the book ›

Sommaire

This precalculus trig graphs lesson 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.

Sine and cosine graphs are periodic waves between -1 and 1 that repeat every 2 pi. In y = a sin(b(x – h)) + k, the amplitude is |a|, the period is 2 pi / |b|, the phase shift is h and the midline is y = k. Tangent has period pi and vertical asymptotes where cosine is zero.

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

Key facts in this precalculus trig graphs lesson

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.

What do the graphs of sine and cosine look like?

The sine and cosine functions are periodic: their graphs repeat every \(2\pi\). Both oscillate between \(-1\) and \(1\) with a smooth wave. The cosine graph starts at its maximum \((0,1)\); the sine graph starts at the midline \((0,0)\) and rises. The cosine graph is the sine graph shifted left \(\frac{\pi}{2}\).

One period of sine (blue) and cosine (orange).

How do you read the equation \(y=a\sin(b(x-h))+k\)?

Constant	Meaning	Formula
\(\|a\|\)	Amplitude (distance from midline to peak)	\(\frac{\text{max}-\text{min}}{2}\)
\(b\)	Controls the period	\(\text{period}=\frac{2\pi}{\|b\|}\)
\(h\)	Phase shift (right if positive)	Start of a cycle
\(k\)	Vertical shift and midline \(y=k\)	\(\frac{\text{max}+\text{min}}{2}\)

A negative \(a\) reflects the graph over the midline. For \(y=3\sin 2x\), the amplitude is \(3\) and the period is \(\frac{2\pi}{2}=\pi\).

\(y=3\sin2x\): maximum \(3\) at \(x=\frac{\pi}{4}\), period \(\pi\).

\(y=\sin\!\left(x-\frac{\pi}{3}\right)\) is the sine graph shifted right \(\frac{\pi}{3}\).

What does the tangent graph look like?

Since \(\tan x=\frac{\sin x}{\cos x}\), tangent is undefined where \(\cos x=0\), at \(x=\frac{\pi}{2}+k\pi\). These are vertical asymptotes. The period of tangent is \(\pi\), the graph passes through the origin, and it increases without bound between asymptotes. For \(y=\tan bx\) the period is \(\frac{\pi}{|b|}\).

Tangent repeats every pi and has vertical asymptotes at odd multiples of pi/2.

How do you build a trigonometric model?

Find the midline \(k=\frac{\text{max}+\text{min}}{2}\) and amplitude \(|a|=\frac{\text{max}-\text{min}}{2}\).
Find the period \(P\) and set \(b=\frac{2\pi}{P}\).
Choose cosine if the model starts at a maximum (or sine if it starts at the midline) to avoid a phase shift.
Check one or two points.

For tides, a depth that ranges from 3 ft to 7 ft with a 12-hour cycle and a high tide at \(t=0\) is \(d(t)=5+2\cos\left(\frac{\pi}{6}t\right)\), because the amplitude is \(2\), the midline is \(5\), and \(b=\frac{2\pi}{12}=\frac{\pi}{6}\).

A tide model: depth in feet against time in hours.

Precalculus trig graphs lesson: questions and answers

What is amplitude?

Amplitude is the distance from the midline to a maximum or minimum. For y = 3 sin x + 2 the amplitude is 3, and the graph goes from -1 to 5.

How do you find the period?

Divide 2 pi by the absolute value of b for sine and cosine, or pi by the absolute value of b for tangent. For y = sin(4x), the period is 2 pi / 4 = pi / 2.

What is a phase shift?

It is a horizontal shift. Write the function as sin(b(x – h)); then h is the shift. For sin(2x – pi) = sin(2(x – pi/2)) the shift is pi/2 to the right.

Why does tangent have asymptotes?

Tangent is sine divided by cosine. Where cosine is 0, the quotient is undefined and the graph shoots off to infinity, creating a vertical asymptote.

How do I choose sine or cosine for a model?

Either works with a phase shift. Cosine is simpler if the data starts at a maximum, and sine is simpler if it starts at the midline going up.

How to use this precalculus trig graphs lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the precalculus trig graphs lesson as a PDF with the button below and keep it with your notes.

All Precalculus chapters: Precalculus math.

Voter.. post

Free printable PDF