The graph of an equation involving and is the set of all points in the cartesian coordinate plane whose coordinates satisfy the equation. The graph of a function is the graph of the equation For example, Figure 1 shows the graph of the equation

We'll use this graph and equation to illustrate some ideas that have much wider applicability. Let's consider making some changes:

Suppose we multiply the value of with a constant. Let's call it . Thus we obtain the new equation

- In this case we increase the value of and so we
*stretch*the graph vertically. -
In this case we decrease the value of and so
*compress*the graph vertically. - We replace with . Geometrically the result is a reflection of the graph of through the -axis.
- In this case the reflection is combined with a vertical compression.
- In this case the reflection is combined with a vertical stretching.

** Figure 2. Several parabolas**

Figure 2 shows the graphs of these equations for

Graphs can be similarly reflected, stretched or compressed in the horizontal direction by multiplying with a constant. However, in the present simple example this effect is equivalent to a vertical rescaling since where .

Consider now the effect of subtracting a constant from :

This is equivalent to

Subtracting a constant from has the same effect in the horizontal direction. Consider the equation

** Figure 3. Translated Parabolas**

Horizontal and vertical shifts can be combined, and any such
combination is called a ** translation**. The general form of a
translation applied to is given by

The **completed square form** of a quadratic function is

It is clear how a general quadratic polynomial

The graph of this particular equation is given in Figure 4. The resulting parabola has and the vertex (. It is obtained from the standard parabola by reflecting it thorough the -axis, stretching it vertically and translating it 1 unit to the left and 8 units up. Note that all of these properties can be immediately read off the equation

** Figure 4. Another Parabola**

The principles illustrated here apply to any equation, so let's restate them:

- A combination of horizontal and vertical shifts is a
**translation**of the graph, a combination of horizontal and vertical compression and stretching is a**scaling**of the graph. - Adding a constant to shifts the graph units to the right if is positive, and to the left if is negative.
- Adding a constant to shifts the graph units up if is positive, and down if is negative.
- Multiplying with a positive constant compresses the graph horizontally if and stretches it horizontally if . If is negative these effects are combined with a reflection through the axis.
- Multiplying with a positive constant compresses the graph vertically if and stretches it vertically if . If is negative these effects are combined with a reflection through the axis.

As usual, I recommend that you do not memorize these facts. Rather think about what's happening in terms of the interplay between the graph and the equation, and figure out the effect of what you are doing, or what you need to do, rather than relying on remembering arcane facts.