Fun With Quadratics
By Brandie Thrasher
Lets begin by examining the graph
y = ax^{2}
It looks as if we have our basic parabola. What
exactly is a parabola?
Well, a parabola is a U-shaped curve for a point known
as the focus and a line (not through the focus) called the directrix. The
parabola is a locus of points, where the distance to the focus is equivalent to
the distance to the directrix.
The standard form for a parabola is
y = a (x – h)^{2} + k
But the general form (that most people recognize) is
y = ax^{2 }+ bx + c
Lets see what happens to our parabola if we change our
value for a (also known as our
coefficient)
Our new parabola (in red) is the equation y = 2x^{2}.
Looks as though this graph is slightly more narrow than or original equation,
but why?
LetÕs use a table and compare potential values
For this table, a = 1
x |
ax^{2} |
y |
-2 |
(1)(-2)^{2} |
4 |
-1 |
(1)(-1)^{2} |
1 |
0 |
(1)(0)^{2} |
0 |
1 |
(1)(1)^{2} |
1 |
2 |
(1)(2)^{2} |
4 |
x |
2x^{2} |
y |
-2 |
(2)(-2)^{2} |
8 |
-1 |
(2)(-1)^{2} |
2 |
0 |
(2)(0)^{2} |
0 |
1 |
(2)(1)^{2} |
2 |
2 |
(2)(2)^{2} |
8 |
With our two tables we can analyze our values for y.
It looks as though the distance between the y values is greater in the second chart
(even as the actual distance increases), than the values in the first chart.
This increase in distance is what causes the second graph to appear
narrower. As our coefficient
increases (is bigger than 1), or distance between values will continue to increase.
How will y = 3x^{2} appear on our graph?
Here it is in blue (y = 3x^{2}), and it is
narrower than the first two. Looks
good!!! But, what about coefficients less than 1 (but greater than zero), how
would they impact our graph?
The green graph represents y = ½ x^{2}
and it appears to e wider than our original equation (in purple). Lets compare
their charts
x |
ax^{2} |
y |
-2 |
(1)(-2)^{2} |
4 |
-1 |
(1)(-1)^{2} |
1 |
0 |
(1)(0)^{2} |
0 |
1 |
(1)(1)^{2} |
1 |
2 |
(1)(2)^{2} |
4 |
x |
½ x^{2} |
y |
-2 |
(1/2)(-2)^{2} |
2 |
-1 |
(1/2)(-1)^{2} |
½ |
0 |
(1/2)(0)^{2} |
0 |
1 |
(1/2)(1)^{2} |
½ |
2 |
(1/2)(2)^{2} |
2 |
In contrast to the previous example, these distances
are less than those of the original equation. As our coefficients decrease
(smaller than 1 but greater than 0) our parabola widens as the distance between
the y values decrease.
Lets look at the graphs of y = ^{1}/_{3 }x^{2}
(in light blue) and y = ¼ x^{2} (in yellow)
So, weÕve looked at coefficients greater than 1 and
others that were greater than zero, but less than 1. What about coefficients
less than zero?
Lets look at y = -2x^{2} and compare it with y
= 2x^{2}
The two graphs appear to have the same shape, except
the grey graph is facing downward, which means that all of its y values are negative.
So we can see that our a value holds the same shape (or difference in distance
between each value) whether negative or positive, but with the coefficient
being negative our y values will all be negative.
As shown here for y = ax^{2} and y = -ax^{2}
Lets see more!
We can now generalize our graphs and know what they
will look like with various coefficients
Reference:
www.mathwords.com/p/parabola.htm