Stephani Eckelkamp

The
Parabola: Friend of Foe?

The graph of a parabola
y=x^{2} has a vertex at (0,0).
The graph is shown below.

Now, let us look at the
graph y=ax^{2}, where a = 1, 2, 3, 10,
½, 1/3, and 1/10É

Éand when a = -1, -2, -3,
-10, -1/2, -1/3, and -1/10.

y = 10x^{2}
and y = -10x

y = 3x^{2} and y = -3x^{2}

y = 2x^{2} and y = -2x^{2}

y = x^{2} and y = -x^{2}

y = ½ x^{2} and y = -1/2x^{2}

y = 1/3x^{2} and y = -1/3x^{2}

y = 1/10x^{2} and y = -1/10x^{2}

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We see that by increasing a the parabola is closing in on the y
axis, and as we decrease a the parabola opens wider towards the x axis. By changing a to a negative integer we
get a reflection in the x axis.

LetÕs investigate sine functions as the variable a.

Here we are looking at
the following equations:

y = sin x^{2} and y = x^{2}

y = sin 2x^{2} and y = 2x^{2}

y = -sin x^{2} and y = -x^{2}

y = -sin 2x^{2} and y = -2x^{2 }

The
graph of the y = ax^{2} equation and the y = sin ax^{2} share
the vertex of (0,0).

An example of the graph to the
right for all values of n from -10 to -.5 and . 5 to 10. Animated graph

y
= nx^{2}

All graphs
have a vertex of (0,0) because we are only working with the equation y = ax^{2}. From my investigations I can conclude
that a (or n in the graph above) changes the width or narrowness of the
parabola. As we can see with the sin
curves above, the wave lengths are closer together the greater a is.