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 x^{2} and (x+a)^{2 }(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 = x^{2 } and y = x^{2 }+ 3 on the same diagram, compare the graphs.

Plot y = x^{2 }− 4, y = x^{2 }− 1

Compare y = x^{2 }and y = x^{2 }+ 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 = x^{2 } and y = (x+ 3)^{2 }on the same diagram, compare the graphs.

Compare y = x^{2 } and y = (x−3)^{2}.^{ }Compare y = x^{2 } and y = (x−4)^{2 }Compare y = x^{2 }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=x^{2} and kx^{2} 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).

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

Umbrella and rain by Luke Walsh on Desmos