​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 $$y= \sin(x) + 1$$:

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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 $$\dfrac13$$.

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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:

$$3x = -270, -90, 90, 270$$​​

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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 $$y = \cos(3x)$$: