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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 = x2 has its minimum at (0,0)

y = x2 + 2 moves it up 2, minimum at (0,2)

y = x2 - 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 = x2 + 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)

Combining the two pieces of information above, from the two different ways of writing the equation, I can tell that this curve must go through (0,3) and (-2,-1). I can now sketch it!

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