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. TranslationsAdding a constant to the functionThe 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, xThe 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 vectorReflections in the x- and y-axesThe 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: 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). Stretches in the x- and y-axesThe 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... 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...