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Transforming trigonometric graphs

Transforming trigonometric graphs

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Explainer Video

Tutor: Labib

Summary

​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)+cy=f(x)+cy=f(x)+c. This will move the graph y=f(x)y=f(x)y=f(x)​ by the vector (0c)\begin{pmatrix} 0 \\ c \end{pmatrix}(0c​). 

Horizontal translations are in the form y=f(x+c)y=f(x+c)y=f(x+c). This will move the graph y=f(x)y=f(x)y=f(x)​ by the vector (−c0)\begin{pmatrix} -c \\ 0 \end{pmatrix}(−c0​).​


Example 1

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


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

Translation=(01)\textit{Translation} = \begin{pmatrix}0\\1 \end{pmatrix}Translation=(01​)​​


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

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


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


Maths; Trigonometry; KS5 Year 12; Transforming trigonometric graphs

​


Stretching

 y=f(cx)y = f(cx)y=f(cx) is a horizontal stretch of the graph y=f(x)y=f(x)y=f(x) with scale factor 1c\dfrac1cc1​.

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


Example 2

Sketch the graph of y=cos⁡(3x)y= \cos(3x)y=cos(3x) between the range −90∘≤x≤90∘-90^{\circ}\le x \le 90^{\circ}−90∘≤x≤90∘.​


Identify the transformation:

Horizontal stretch of the graph y=cos⁡(x)y=\cos(x)y=cos(x) with scale factor 13\dfrac1331​.

​​

Before identifying where y=cos⁡(x)y=\cos(x)y=cos(x) crosses the xxx-axis, adjust the range of xxx values in the question by a multiple of 333​: 

−270≤3x≤270 \\ -270 \le 3x \le 270 −270≤3x≤270​​


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

3x=−270,−90,90,2703x = -270, -90, 90, 2703x=−270,−90,90,270​​

​

Adjust these values by a scale factor of 13\dfrac1331​ to find where y=cos⁡(3x)y = \cos(3x)y=cos(3x) crosses the xxx-axis:

x=−90,−30,30,90x = -90, -30, 30, 90x=−90,−30,30,90


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

Maths; Trigonometry; KS5 Year 12; Transforming trigonometric graphs




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Trigonometric graphs - Higher

Unit 1

Trigonometric graphs - Higher

Trig graphs: sin x, cos x, tan x

Unit 2

Trig graphs: sin x, cos x, tan x

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Transforming trigonometric graphs

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FAQs - Frequently Asked Questions

What transformation is y=a*tan(x) to the graph y=tan(x)?

A vertical stretch by the scale factor a.

What transformation is y=cos(x)-a to the graph y=cos(x)?

A vertical translation moving y values by -a.

What transformation is y=sin(ax) to the graph y=sin(x)?

A horizontal stretch by the scale factor 1\a. ​

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