Graph sine, cosine, and tangent functions

The learner understands how sine, cosine, and tangent graphs arise from their unit-circle values and how amplitude, period, phase shift, and vertical translation determine the shape and position of $y=a\sin(bx+c)+d$ and $y=a\cos(bx+c)+d$ . They interpret key points, zeros, maxima and minima, periodicity, and, for tangent, repeating branches and vertical asymptotes in radians, recognizing that tangent has no amplitude; inverse trigonometric functions, parametric or polar graphs, and more advanced transformations are not included.

Detailed Explanation: Graph sine, cosine, and tangent functions

A sine or cosine graph can be built from the unit-circle values

0,\ 1,\ 0,\ -1,\ 0

over one cycle. Transformations change the graph’s height, width, and position.

For

y=a\sin(bx+c)+d \quad\text{or}\quad y=a\cos(bx+c)+d,

Amplitude: $\vert a \vert$
Period: $\dfrac{2\pi}{ \vert b \vert }$
Phase shift: $\displaystyle -\frac{c}{b}$
Vertical translation: $d$

The midline is $y=d$ . If $a<0$ , the graph is reflected across its midline.

Worked example

Graph

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

1. Identify the transformations

Compare the equation with

y=a\sin(bx+c)+d.

Here,

a=2,\qquad b=1,\qquad c=-\frac{\pi}{3},\qquad d=1.

Therefore:

Amplitude: $\vert 2 \vert =2$
Period:

\frac{2\pi}{1}=2\pi

Phase shift: $\displaystyle \frac{\pi}{3}$ units to the right
Midline: $y=1$

The maximum value is

1+2=3,

and the minimum value is

1-2=-1.

2. Divide one period into four equal parts

A sine graph uses five key points over one period:

0,\quad \frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi.

Since the graph is shifted right by $\frac{\pi}{3}$ , add $\frac{\pi}{3}$ to each $x$ -coordinate:

\begin{array}{c|ccccc} x & \frac{\pi}{3} & \frac{5\pi}{6} & \frac{4\pi}{3} & \frac{11\pi}{6} & \frac{7\pi}{3}\\ \hline y & 1 & 3 & 1 & -1 & 1 \end{array}

The $y$ -values come from the usual sine pattern, stretched by $2$ and shifted up $1$ .

3. Plot and connect the points

Plot

\left(\frac{\pi}{3},1\right),\quad \left(\frac{5\pi}{6},3\right),\quad \left(\frac{4\pi}{3},1\right),\quad \left(\frac{11\pi}{6},-1\right),\quad \left(\frac{7\pi}{3},1\right).

Join them with a smooth sine curve. Then repeat the same pattern every $2\pi$ units.

The graph has:

Midline $y=1$
Maximum $y=3$
Minimum $y=-1$
Period $2\pi$

For a cosine graph, use the same process, but the basic key points begin at a maximum:

1,\ 0,\ -1,\ 0,\ 1.

For tangent,

y=a\tan(bx+c)+d,

there is no amplitude. Its period is

\frac{\pi}{|b|},

and its vertical asymptotes occur where

bx+c=\frac{\pi}{2}+k\pi,\qquad k\in\mathbb Z.

Between the asymptotes, tangent branches pass through the midline $y=d$ and repeat every period.

Learn by doing: Graph sine, cosine, and tangent functions

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Sinusoidal Function Parameters (4 Params) - Graph to Function

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