Transforming trigonometric graphs

In a nutshell

Trigonometric graphs can undergo a variety of transformations. You will need to know how to apply translation and stretches to different trigonometric graphs.

Translations

Vertical translations are in the form $y=f(x)+c$ . This will move the graph $y=f(x)$ by the vector $\begin{pmatrix} 0 \\ c \end{pmatrix}$ .

Horizontal translations are in the form $y=f(x+c)$ . This will move the graph $y=f(x)$ by the vector $\begin{pmatrix} -c \\ 0 \end{pmatrix}$ .

Example 1

Sketch the graph of $y= \sin(x)+1$ in the range $0\le x \le 360^{\circ}$ .

Identify the transformation of the curve $y= \sin(x)$ to $y = \sin(x) + 1$ :

$\textit{Translation} = \begin{pmatrix}0\\1 \end{pmatrix}$

Find the maximum and minimum of $y = \sin(x)+1$ :

$\begin{aligned} \textit{Maximum} &= &1&+1 &= 2 \\ \textit{Minimum} &= &-1&+1 &= 0 \end{aligned}$

Sketch the graph of $y= \sin(x) + 1$ :

Maths; Trigonometry; KS5 Year 12; Transforming trigonometric graphs

Stretching

$y = f(cx)$ is a horizontal stretch of the graph $y=f(x)$ with scale factor $\dfrac1c$ .

$y=cf(x)$ is a vertical stretch of the graph $y=f(x)$ with scale factor $c$ .

Example 2

Sketch the graph of $y= \cos(3x)$ between the range $-90^{\circ}\le x \le 90^{\circ}$ .

Identify the transformation:

Horizontal stretch of the graph $y=\cos(x)$ with scale factor $\dfrac13$ .

Before identifying where $y=\cos(x)$ crosses the $x$ -axis, adjust the range of $x$ values in the question by a multiple of $3$ :

$\\ -270 \le 3x \le 270$ 

Identify where the curve of $y=\cos(x)$ crosses the $x$ -axis in this range:

$3x = -270, -90, 90, 270$ 

Adjust these values by a scale factor of $\dfrac13$ to find where $y = \cos(3x)$ crosses the $x$ -axis:

$x = -90, -30, 30, 90$

Sketch $y = \cos(3x)$ :

Maths; Trigonometry; KS5 Year 12; Transforming trigonometric graphs