by

Jane Yun

With many available technologies, it has become
possible to construct graphs following equation.

*a***x ^{2} + bx + c = 0**

Also, it can overlay several graphs of

**y = ax^{2}
+ bx + c**

for different values of a, b, or c as the other
two are held constant. From these
graphs, we can discuss the patterns for the roots of

*a***x ^{2} + bx + c = 0**

**Investigation
1. Changing Ō bÕ**

LetÕs start the investigation when b = -3, -2,
-1, 0, 1, 2, 3 for the following quadratic equation.

**y = x ^{2} + bx
+ 1**

From observing the graph above, the shape of
graph is a parabola. The parabola
shifts to right or left as *b* changes
and always passes through the same point on the y-axis at (0,1). For b < -2, the parabola passes
through the x-axis twice with positive x values; that is, the equation has two
positive real roots. For *b* > 2,
the parabola passes through the x-axis twice with negative x values. This indicates that the equation has
two negative real roots. For *b* = -2, the parabola is tangent to the
x–axis, so the original equation has one real positive root at the point
of tangency. Again, for b = 2, the
parabola is tangent to the x-axis.
Thus, the original equation has one real negative root at the point of
tangency. For -2 < *b* < 2, the parabola does not pass
through the x-axis. This means
that the equation has no real roots.

**Investigation
2. The xb- plane**

LetÕs consider the following equation.

**x ^{2} + bx
+ 1 = 0**

By setting *b*
= y, you get the following graph.

Notice that there is a vertical asymptote at x =
0. When b = 2, the graph is
tangent to y = 2, so this equation has one real negative root at the point of
tangency. Again, when b = -2, the graph
is tangent to y = -2. Thus, this
equation has one real positive root at the point of tangency.

Now, if we take any particular value of *b*, such as *b* = 5, and overlay this equation on the graph, we add a line that
is parallel to the x-axis. If *b* = 5
passes through the curve in the xb plane, the intersection points aligns to the
roots of the original equation for that value of *b*. Look at the graph
below.

For each value of b, we get a horizontal
line. By looking at the graph
above, it clearly shows that we get two negative roots of the original equation
when b > 2, one negative real root when b = 2, one positive real root when b
= -2, no real roots for -2 < b < 2, and two positive real roots when b
< -2.

**Investigation
3. When c = -1**

LetÕs consider following equation.

**x ^{2} + bx
- 1 = 0**

Notice that there is still a vertical asymptote
at x = 0 and the shape of graph is a hyperbola.

If
we take any particular value of *b*,
such as *b* = 5 again, and overlay this equation on the graph,
we add a line that is parallel to the x-axis. If *b* = 5 passes through the curve in the xb plane, the intersection
points aligns to the roots of the original equation for that value of *b*.
It is clear that, for
each value of *b*, we will get a
horizontal line and intersects the graph of original equation, **x ^{2} + bx
- 1 = 0**

LetÕs explore more by varying the values of c.

By looking at the graph
above, the equation, x^{2} + bx + c = 0, always has one real root when
c = 0. For c > 0, the equation
can have among two real roots, one root, or no root. When c < 0, the equation always have two real roots –
one positive and one negative.