Transformations of functions

What this area is about

Most graphs you meet are a few basic shapes slid, flipped or stretched. Each change to a rule matches a change to its graph. For k > 0, adding k to the output slides the graph up k units; for h > 0, replacing x by x - h slides it right h units. Negating the whole output flips the graph upside down, and multiplying the output by a > 1 stretches it away from the horizontal axis. This area reads such moves off a rule, draws the result, and runs the process backward.

How these fit together

The shapes. Parent functions is an area of its own: the plainest graph in each family, such as y = x^2, \displaystyle y = \sqrt{x} and \displaystyle y = \frac{1}{x}.

Single moves. Shifting a graph slides it up, down, left or right. Reflecting across the axes turns it upside down or makes it face the other way. Stretches and compressions multiply the output or the input by a number.

Procedures. Combining transformations reads several moves off one rule, in the order they happen. Graphing by transforming a parent graph draws the curve and finds its domain and range. Finding the rule of a transformed graph goes from a curve or table back to the rule.

A path through it

Shifts, then reflections, then stretches, then combining them. Graphing by transformations needs those and the parent functions; finding the rule from a graph comes last.

Next, the same moves draw exponential, absolute value and square root graphs, and the trigonometric graphs.