Quadratic Questions

LetÕs observe how the parabola y=ax^{2 }changes
as the value of *a*
is varied.

First, we graph y=x^{2}, or y=(1)x^{2}^{.}

^{ }

This is the simplest parabola to graph. We see that
the y and x intercepts are both the origin. Since the function is quadratic,
there will be no negative y values. Therefore, f(x)
and f(-x) both equal the same y value.

Now, letÕs observe what happens as we vary the value
of *a*.

*a**=0*

We see that when a=0 y=(0) x^{2}, or y=0,
which is the x-axis. No matter what our input vale for x is, the output value
will be 0.

*a**>1*

We see that when our *a* value is greater than 1 the
parabola appears to be ÒstretchedÓ. This is because for every input value of x,
the output value is being multiplied by the value of *a*. Since *a*
is greater than 1 in this case, the output value will be larger than the
output value of y=x^{2}. Therefore, when *a* is greater than 1, the function
is increasing at a faster rate than y=x^{2}.

*0<a<1*

We see that when our *a* value is greater than 0 and
less than 1 the parabola appears to be ÒshrunkenÓ. This is because for every
input value of x, the output value is being multiplied by the value of *a*. Since *a* is between 0 and 1 in this
case, the output value will be smaller than the output value of y=x^{2}.
Therefore, when *a*
is between 0 and 1, the function is increasing at a slower rate than y=x^{2}.

*Negative a*

We see that when our *a* value is negative the parabola
is reflected over the x-axis. It does not matter whether the input value is
positive or negative. It will become positive when it is squared, then become
negative when it is multiplied by a negative value of *a*. Therefore, when *a* is
negative, all of the output values of the function are negative, as well. We
see that the same properties as above hold as well. If *0>a>-1*, the parabola appears to be shrunken, and if *a<-1* the parabola appears to be
stretched.

*Conclusion*

In conclusion, we see that the value of *a* can greatly affect even the most basic
quadratic functions. It can affect the rate at which the function increases or
decreases, and the orientation of the parabola.