__
Transformations of graphs__

You probably already know that shapes can
be undergo the following transformations...

Translation

Reflection

Rotation

Enlargement

__Of course, graphs can also undergo
these transformations but there are a few new ones you need to know too__

__1) Translation__

A translation is when the shape of the
graph remains identical but its position on the graph changes

y = x^{2} has its minimum at
(0,0)

y = x^{2} + 2 moves it up 2,
minimum at (0,2)

y = x^{2} - 2 moves it down 2,
minimum at (0,-2)

y = (x**+2**)^{2} moves it **
left** 2, minimum (**-2**,0)
*remember, adding a positive value to x moves it the
"wrong" way*

y = (x**-2**)^{2} moves it **
right** 2, minimum (**2**,0)

**2**__)
__**Stretches**

You may be familiar with the curve y = sin
x. If not, it is drawn out on the
Trigonometric functions page. It's maximum y value is +1 and its minimum y
value is -1.

y = 2(sin x) means it now reaches 2 and -2
Stretch scale factor 2 in the direction of the y
axis

y = ^{1}/_{2}(sin x) means
it now only reaches 0.5 and -0.5
Stretch scale factor ^{1}/_{2} in the direction of the y axis

y = sin(2x) does not change the y maxima
and minima but **'squashes'** it in the x direction. It now crosses the x
axis at 0, 90, 180, 270, 360...
Stretch scale factor ^{1}/_{2} in
the direction of the x axis

y = sin(2x) does not change the y maxima
and minima but **stretches** it in the x direction. Just as with
translations, this is the opposite of what you might expect. It now crosses the
x axis at 0 and 360 (and nowhere in-between)
Stretch scale factor **2** in the direction of the x
axis

__3) Reflections__

y = sin(-x) is a reflection in the **
y** axis *(opposite to what you might expect)*

y = -(sin x) means all the y values are
multiplied by -1 so this is a reflection in the **x**
axis *(opposite to what you might expect)*

__4) Enlargements__

An enlargement scale factor e.g. 7 is a
stretch scale factor 7 in the direction of the x axis followed by a stretch
scale factor 7 in the direction of the y axis (or the other way round)

__Function notation__

A function is the 'proper' word for the
equation of a particular line. y = 2x + 1 is a function as is y = sin x. The
first function multiplies x by 2 and adds 1 to get y, the second takes the sine
of x to get y. But sometimes (to make transformation questions harder!) they
will hide precisely what the function does to x to get y. They will simply write
'y = f(x)' to represent a function.

Of course, exactly the same rules apply as
before. y = f(x) + 4 is a translation of (0,4) [zero across and 4 up] and y =
f(0.5x) is a stretch scale factor 2 in the direction of the x axis.

__Sketching quadratics__

A quadratic curve has the equation y = x^{2}
+ 4x + 3

From this information you can tell that it
crosses the y axis at y = 3

Next you have to complete the square (see
the Algebra
section)

This gives y = (x + 2)^{2} -1

From this information you can tell that
its minimum point has been 'translated' from (0,0) by (-2,-1) so the minimum
point of the graph is (-2,-1)