Graphing Using Translations

In the section on translations we've seen how the graphs of f(x + h) and f(x - h) are related to y = f(x). If h>0 then f(x+h) is a translation of f(x) h units to the left while f(x-h) is a translation of f(x) h units to the right. We will review this and other translations using the basic quadratic function,

Example 1

Write out the formulas for f(x+2) and f(x-2) when f(x) is the squaring function. Graph f(x), f(x+2), and f(x-2) on the same coordinate system.

Solution .

Because of the definition of f(x), and

Now make the required graph.

Note that the vertex on f(x+2) is (-2, 0) while the vertex of f(x-2) is (2, 0)

We now combine the horizontal translation with a vertical one.

Example 2

Sketch Specify the vertex and the axis of symmetry.

Solution .

Example 3

Now graph and note the vertex and axis of symmetry.

Solution .

Note that this parabola opens down.

The standard form ( or vertex form) of a quadratic function is

where the vertex is V=(h, k) and the axis of symmetry is x = h.

When a>0, the graph opens upward and k is the smallest y value on the graph.

When a<0, the graph opens downward and k is the largest y value on the graph.