This problem follows on from Quadratic
Transformations.

If you don't have access to a
computer room, begin the lesson by asking learners each
to select a number, and to perform the following operations,
and write down their answer (perhaps on a mini-whiteboard)

Square your number. Multiply your initial number by three. Add
these two answers. Subtract 4.

Then ask them to perform the following operations using the
same starting number:

Square your number. Multiply your initial number by three. Add
these two answers. Subtract 4. Multiply the answer by
two.

Display a coordinate grid on the board at the front, and
choose some students to plot points, the $x$ coordinate being the
number they chose, and the $y$ coordinates being their answers -
it's a good idea to plot these in different colours on the same
graph.

If you have a computer and projector, display the Number
Plumber and ask learners to suggest inputs and work out what the
outputs will be.

If you do have access to a
computer room, begin the lesson by allowing learners to
explore the Number Plumber interactivity in pairs. Click here to see how to
use the NRICH Number Plumber, or click here to see a
video for teachers wishing to create their own examples.

Once everyone has had a chance to see what's going on, give
them time in pairs to discuss the following questions:

What happens when you input the same number into both
functions?

How are the two functions related?

How are the graphs related?

Encourage learners to suggest conjectures about relationships
between graphs, and perhaps introduce the notation $y=f(x)$ in
order to express ideas about the relationship between functions.
Software such as the free-to-download Geogebra may be useful for
plotting other graphs in order to explore related functions of the
form $y=f(x)$ and $y=af(x)$, or alternatively learners can be
encouraged to work in small groups plotting related graphs on graph
paper and sharing their results. As learners begin to notice
relationships between the graphs, encourage them to explain why the graphs are linked in this
way by relating the differences in the graphs to the differences
between the two function machines.

After tackling the relationship between $y=f(x)$ and
$y=af(x)$, the second Number Plumber interactivity offers a chance
to explore the relationship between $y=f(x)$ and $y=f(ax)$ - this
can be approached in a very similar way. For those exploring
without computers, the two functions are:

$x^2+3x-4$ and $(2x)^2+3(2x)-4$

How are the two functions related?

How are their graphs related?

What is the relationship between the graph $y=f(x)$ and the graph
$y=af(x)$?What is the relationship between the graph $y=f(x)$ and the graph $y=f(ax)$?

Set a challenge by choosing two quadratic functions and asking
learners to devise a method for transforming one function into
another using stretches and translations.

Learners can apply their understanding of transformation of
functions by working on the challenges Parabolic
Patterns, More Parabolic
Patterns, and Parabolas
Again.