Transformations of Graphs

Once you know the graph of y = f(x), you can build a whole family of related graphs by moving and stretching it — without re-plotting a single point. There are four standard moves.

y = f(x) + a shifts the curve up by a.
y = f(x - a) shifts the curve right by a — note the minus sign moves it the positive way.
y = a \cdot f(x) stretches the curve vertically by a factor of a.
y = -f(x) reflects the curve in the x-axis.

Here is the rule that ties them together: a change inside the bracket affects x and goes the opposite way to what you expect, while a change outside the bracket affects y and does the obvious thing.

Starting from the graph of y = f(x):

+\,a outside the bracket → shift up by a;
(x - a) inside the bracket → shift right by a;
\times a outside the bracket → vertical stretch by factor a;
a minus outside the bracket → reflect in the x-axis.

Shift it yourself

The faint dashed curve is the base parabola y = x^2. Pull the sliders to move it: h slides it right (inside the bracket) and k slides it up (outside). Together they make y = (x - h)^2 + k.