Trigonometric Graphs & Transformations

Trigonometric Graphs & Transformations

Sketching a graph and how to transform them

Basic Trigonometric Graphs

•  $y=\sin x\$  

•  $y=\cos\ x$  

•  $y=\tan x$  

This is a graph of  $y=\sin x$  for  $-90\degree\le x\le360\degree$  

What about the others? How do they look like?

Your turn

Try all equations you can think of.

Enter the equations in the following website, and explore.

Don't forget to click on autoscale in settings.

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Multiple Choice

When $y=\sin x$  is transformed into  $y=2+\sin\ x$ , what happens to the graph?

• Graph is shifted downwards by 2 units

• Graph is stretched by 2 units, parallel to the y-axis

• Graph is shifted upwards by 2 units

• Graph is stretched by 2 units, parallel to the x-axis

Try again with Autograph

Explore what happens to the graph when you add or minus a constant.

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Multiple Choice

When  $y=\sin x$  is transformed into $y=-\sin\ x$ , what happens to the graph?

• Graph is reflected in the y-axis

• Graph is shifted upwards by 1 unit

• Graph is reflected in the x-axis

• Graph is stretched by a factor of 2, parallel to the x-axis

Try again with Autograph

Explore what happens to the graph when you it becomes negative.

Multiple Choice

When $y=\sin x$  is transformed into $y=2\sin\ x$ , what happens to the graph?

• Graph is stretched by a factor of  $\frac{1}{2}$  , parallel to the y-axis

• Graph is stretched by a factor of  $\frac{1}{2}$  , parallel to the x-axis

• Graph is stretched by a factor of 2, parallel to the y-axis

• Graph is stretched by a factor of 2, parallel to the x-axis

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Multiple Choice

When  $y=\tan x$  is transformed into $y=\tan2\ x$ , what happens to the graph?

• Graph is stretched by a factor of  $\frac{1}{2}$  , parallel to the y-axis

• Graph is stretched by a factor of  $\frac{1}{2}$  , parallel to the x-axis

• Graph is stretched by a factor of 2, parallel to the y-axis

• Graph is stretched by a factor of 2, parallel to the x-axis

Try again with Autograph

Explore what happens to the graph when you multiply the function or the x in the function with a factor e.g. $y=\cos\ \frac{1}{2}x$   

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Multiple Choice

When  $y=\sin x$  is transformed into   $y=\sin\ \left(x+30\right)\degree$ , what happens to the graph?

• Graph is shifted by  $30\degree$  to the left

• Graph is shifted by  $30\degree$  to the right

• Graph is shifted by  $30\degree$   upwards

• Graph is shifted by  $30\degree$  downwards

Try again with Autograph

Explore what happens to the graph when you add or minus an angle from the x function e.g. $y=\tan\ \left(x-45\right)\degree$   

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In summary:

•  $y=f\left(x\right)+a$  : Graph shifts upwards (downwards when minus) by a units.

•  $y=af\left(x\right)$  : Graph stretches by factor a, parallel to y-axis

•  $y=f\left(ax\right)$  : Graph stretches by a factor of  $\frac{1}{a}$  , parallel to the x-axis

•  $y=f\left(x+c\degree\right)$  : Graph shifts to the left (right when minus) by  $c\degree$ 

•  $y=-f\left(x\right)$  : Graph is reflected in the y-axis.

Take a look at this image carefully.

Take note of where the curve cuts the x-axis and y-axis. It is also helpful to note the maximum and minimum values.

The next slide, will require you to answer from this graph.

Multiple Choice

What is the equation of this graph?

• $y=3\sin\left(x+60\right)\degree$

• $y=3\cos\left(2x+60\right)\degree$

• $y=2\sin\left(3x+60\right)\degree$

• $y=3\sin\left(2x-60\right)\ \degree$

Homework!

Take a screenshot of a trigonometric equation you tried on Autograph.

Be creative.

Make it as complicated as you want.

Send the screenshot to our groupchat when you're done.