Assignment # 2

By Sharren M. Thomas

**In this exploration I will
examine the graph y = ax ^{2} on the same axes for different values of
a.**

For a =
1; y = x^{2} is a parabola
whose vertex is at the origin. See the graph below.

For a = 2; y = 2x^{2 }is a parabola whose
vertex is at the origin, but the graph of the parabola is now stretched
vertically by a factor of 2, each y-coordinate at each point has been
multiplied by 2. See both graphs below.

For a =
1/4 y = (1/4) x^{2} is a parabola
whose vertex is at the origin, but the graph of the

parabola
is now compressed vertically, each y – coordinate at each point

has
been multiplied by 1/4. See both graphs below.

Conclusion:

For y = **a**x^{2} with **a**> 1 the graphs will be parabolas whose vertices are
at the origin that are stretched vertically by a factor of **a.** Click the movie below. Notice that the
parabolas are becoming more narrow or sharper at the vertices.

For y = **a**x^{2 }with 0 < **a** > 1 the graphs will be parabolas whose vertices
are at the origin that are compressed vertically by a factor of **a**. Click the movie below. Notice that the
parabolas are becoming wider or flatter at the vertices.

For a =
-1; y = -x^{2 }is a parabola whose
vertex is at the origin, but the graph of the parabola has been reflected
across the x-axis. Each y-coordinate of the new graph has been multiplied
by -1. See the graph below.

For y = **a**x^{2 }with a < 0; the graphs will be
parabolas whose vertices are at the origin that are reflected across the
x-axis. When **a** is a negative
fraction it the graph is both reflected across the x-axis and compressed
vertically by a factor of **a**.
Click the movie below. Notice that the parabolas are becoming wider or
flatter at the vertices.

For y = **a**x^{2 }with a <0; the graphs will be parabolas whose vertices are at
the origin that are reflected across the x-axis. When **a **is a negative integer the graph is both reflected
across the x-axis and stretched vertically by a factor of **a**. Click the movie below. Notice that the
parabolas are becoming narrow or sharper at the vertices.