Graphs and Transformations


Note how easy it is using the Desmos graphing calculator to show a graph and then the same graph after a transformation. For example see here the graph of x2 and (x+a)(or click on  the image).

As you can see all you need do is define f(x) which then gives you the ability to graph transformations of f(x). Using a slider means students can explore what happens if the value of a changes. In this particular example a is restricted to integer values from 0 to 3 but this is very easy to change by editing the slider; simply click on the numbers at the ends of the slider and choose the values required, you can also choose the step size.

I like the ability to add text to graph pages which means it is possible to add a few words of explanation or even questions for students.

I have used these prompts before in a computer room, the aim being for students to explore and generalize. The headings are links to Desmos graph pages students could experiment with.

Transformations Exercise

For linear, quadratic and the sine and cosine functions the following transformations should be understood:

1. y=f(x)+a quadratic

y=f(x)+a trigonometric
Eg plot y = x2   and y = x+ 3   on the same diagram, compare the graphs.
Plot y = x− 4, y = x− 1
Compare y = x2  and y = x+ k where k is any integer, positive or negative.
Also compare y = sin x and y = sin x + 3.
Compare y = sin x and y = sin x + k where k is any number

2. y = f(x+a) quadratic

 y = f(x+a) trigonometric
Eg plot  y = x2   and y = (x+ 3)on the same diagram, compare the graphs.
Compare y = x2   and y = (x−3)2.  Compare y = x2   and y = (x−4)2
Compare y = x2  and y =(x+ k)2 where k is any integer, positive or negative.
Compare y = sin x and y = sin (x+90) , sin (x – 90), sin(x+180)
Compare y = cos x with y = cos(x+90), cos (x-90)

3. y = af(x) trigonometric
Compare y = sinx and y = 2sinx.
Compare y = cosx and y = 3cosx and y =  0.5cos x
Compare y=x2 and kx2 where k is any integer positive or negative.

4. y = f(ax) trigonometric
Compare y = sinx and y = sin2x. Compare y = sin x and y = sin(0.5x).
Compare y = cos x and y = cos 3x. Compare y = cos x and y = cos(0.5x)

See this page for more information on the Demos graphing calculator.

Update – I must add these reactions from Twitter after publishing this post, from @Desmos (click on the image for a great example on transformations of points).
Desmos - transformations

and this fabulous Umbrella and Rain from Luke Walsh, try the slider and watch the rain and umbrella!
Luke Walsh tweet

Umbrella and Rain

Umbrella and rain by Luke Walsh on Desmos

2 comments on “Graphs and Transformations

  1. This is very similar to the way I introduced it to Year 10. A couple of ICT based lessons with Desmos then followed by some whiteboard work based on shifting stretching flipping etc exploring the difference between ‘inside’ and ‘outside’ the function bracket.

    The use of sliders is much more powerful than a series of static images.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s