University of Georgia

It has now become a rather standard exercise, with availble technology, to construct graphs to consider the equation

and to overlay several graphs of

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

can be followed.

**Investigation1 **through**
**grapghs in the xy plane

Let's consider the graphs of

for b = -3, -2, -1, 0, 1, 2, 3.

1) The parabola always passes through the same point (0,1) on the y-axis

2) For b < -2

the parabola will intersect the x-axis in two points with positive x values

(i.e. the original equation will have two real roots, both positive)

3) For b = -2

the parabola is tangent to the x-axis and so the original equation has one real and positive root(i.e. x=1) at the point of tangency

4) For -2 < b < 2

the parabola does not intersect the x-axis -- the original equation has no real roots

5) For b = 2

the parabola is tangent to the x-axis (one real negative root x=-1)

6) For b > 2

the parabola will intersect the x-axis in two points with negative x values

(i.e. the original equation will have two real roots, both negative)

Further through an exploration

we can show that the locus of vertice is the parabola

**Investigation2 **through**
**grapghs in the xb plane

Let's consider graphing this relation

We get the following graph

If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis.

If the line b=some value intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b.

For each value of b we select

1) For b > 2

we get two negative real roots of the original equation

2) For b = 2

we get one negative real root

3) For -2 < b < 2

we get no real roots

4) For b = -2

we get one positive real root

5) For b < -2

we get two positive real roots

**Investigation3 **through**
**grapghs in the xc plane

Let's consider graphing this relation

We get the following graph

If we take any particular value of c, say c = 3, and overlay this equation on the graph we add a line parallel to the x-axis.

For each value of c we select,

its graph will be a line crossing the parabola in **0, 1,
or 2 points -- the intersections** **being
at the roots of the orignal equation** at that value
of c

1) For c > 4

we get no real roots

2) For c = 4

we get one negative real root

3) For 0 < c < 4

we get two negative real roots

4) For c = 0

we get 0 and -4 as the roots

5) For c < 0

we get one negative real root and one positive root

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