Transformations

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Transformations

Now that we know how to extract basic information from a function and sketch its curve graphically, we can consider how to alter the function to translate, reflect or stretch the graph.

Adding a constant to the function

The graph of f(x) + k (where k is a constant), has the same shape of the function f(x) but is k units higher as the diagram shows:

Transformations

Therefore f(x) + k is a translation of f(x) by the vector

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Adding a constant to the variable, x

The graph of f(x + a) is also the same shape as f(x), but this time it moves the graph 'a' units to the left (i.e. the graph happens 'a' units earlier). Taking the example above gives:

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Therefore f(x + a) is a translation of f(x) by the vector

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Using both rules:

The graph similar to y = x2 that turns at the point (2, 5) would have equation y = (x − 2)2 + 5; and in general,

f(x) = f(x − a) + b is a translation of f(x) by the vector

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The graph of f(-x) is simply a reflection of f(x) in the y-axis. This is because the values of x that were positive are now negative and the previous negative values of x are positive. This swap is the same as reflecting in the y-axis.

For example:

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The graph of - f(x) is a reflection of f(x) in the x-axis. If this scenario the y values swap (what was positive becomes negative and vice versa).

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The graph of af(x) is a stretch scale factor a in the y-axis. This is because all the y-values become 'a' times bigger.

The graph of f(ax) is also a stretch. This time the multiple affects the x-values. (Everything happens 'a' times quicker.) Therefore:

The graph of f(ax) is a stretch scale factor 1/a in the x-axis.

Example:

The graph of f(x) = 2 sin 2x is a stretch of sin x by scale factor 2 in the y-axis, and a stretch scale factor 1/a in the x-axis.

f(x) = sin x The original function...

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f(x) = 2 sin x Stretches the original function in the y axis...

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f(x) = 2 sin 2x Compresses (the inverse of stretches) the 2 sin x function in the x-axis...

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