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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:
Therefore f(x) + k is a translation of f(x) by the vector
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:
Therefore f(x + a) is a translation of f(x) by the vector
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
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.
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).
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.
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...|
|f(x) = 2 sin x||Stretches the original function in the y axis...|
|f(x) = 2 sin 2x||Compresses (the inverse of stretches) the 2 sin x function in the x-axis...|
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