**P a r a b o l a
**

**By**

**Jane Yun**

**
**

**We know how
to graph the quadratic equation y = x^{2}. Then, we will explore what is changing when we change the
values of Ôa Ô.**

**
**

Figure
1

**Figure 1
shows the graph of four functions of the form y = ax^{2} when a > 0, **

*a* = ¼, *a* = ½, *a* = 1, *a* = 3, and *a* =10.

**Notice that the larger
the value of a, the ÒnarrowerÓ the graph, and the smaller the value of a,
the ÒwiderÓ the graph.
**

Figure
2

**The next
Figure 2 shows the graphs of y = ax^{2} when a < 0. **

*a* = - 1/4, *a* = -1/2, a = -1, *a* = -3, *a* = -10.

**Notice that
these graphs are reflections about the x-axis of the graphs in Figure 1.**

**Based on the
results of these two figures, we can draw some general conclusions about the
graph of y = ax^{2}. First,
as │a│ increases, the
graph becomes narrow (a vertical
stretch), and as │a│ gets
closer to zero, the graph gets wider
(a vertical compression). Second, if a is positive, then the graph opens up, and if ÔaÕ is negative, the graph opens down.
**

**
**

Figure
3

**The graphs
in Figures 1 & 2 are typical of the graphs of all quadratic equations,
which we call parabolas. Refer to Figure 3, where two parabolas
are pictured. The one on the left
opens up and has a lowest point; the one on the right opens down and has a
highest point. The lowest or
highest point of a parabola is called the vertex. The vertical line passing through the vertex in each
parabola is called the axis of symmetry of the parabola. **