Assignment 3
# Quadratic and Cubic
Equations

### Investigation 5

### Graphs on the xa Plane

We will now consider the equation
, and
by graphing it in the xa plane we get the following graph.

If we take any particular value
of a, say a = -0.5, and overlay this equation on the graph we
add a line parallel to the x-axis. If it intersects the curve
in the xa plane the intersection points correspond to the roots
of the original equation for that value of a. We have the following
graph

For each value of a we select,
we get a horizontal line. It is clear on a single graph that we
get two real roots of opposite signs, for the original equation
when a < 0, one positive real root when 1/4 > a > 0,
and no real roots for a > 1/4.

What would happen if we let b=-1
instead of b=1. We would get the following graph

If we keep a and c constant the
graph changes a great deal as we change b. We can see in the graph
below a few different values of b. The purple graph shows b=(-3),
the red graph shows b=(-2), the blue graph shows b=(-1), the green
graph shows b=0, the light blue graph shows b=1, the yellow graph
shows b=2, and the grey graph shows b=3.

Or you can click HERE
to see the animation of the graph as b changes.

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