# Mathematics 1010 online

## Parabolas

### The Graph of .

Figure 1. A parabola

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

This particular graph is an example of a parabola. Note that it is symmetric with respect to the -axis which is called its axis or line of symmetry. The lowest point on the parabola (which in this case is the origin) is called its vertex.

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

### Rescaling

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

If this is the same equation as before. Let's consider some other possibilities, however:

• 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 .

### Translations

Consider now the effect of subtracting a constant from :

This is equivalent to

and so it has the effect of raising the graph of by units. Of course, if is negative, the graph is lowered (by units).

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

and compare it with the original equation . As in , is never negative. It assumes its minimum value () at . Thus the graph of is that of shifted units to the right. If is negative, the graph is shifted units to the left.

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

which is often written as

A translation does not change the shape or orientation of a graph. It only changes its location. Figure 3 shows translations of the graph of with

### The Completed Square Form

The completed square form of a quadratic function is

Its graph is a parabola whose vertex is . If it opens up, if it opens down. (If the graph is a horizontal line which you can think of a s a degenerate parabola.) The vertical line is the axis or line of symmetry of the parabola.

It is clear how a general quadratic polynomial

can be converted to the standard form. Just complete the square, as illustrated in the following example:

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

### Scaling and Translating Graphs

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.