**Investigation of Quadratics in the xb
plane**

**by**

**Molade Osibodu**

**Objective:**** **To
show how to determine the roots of a quadrtric function by varying* b***.**

Recall that the standard quadratic equation is wriiten in the form . Now consider the equation ax²+ bx + c where a = 1 and c = 1.

To determine the roots of the equation, it has
to equal 0. Therefore, the equation we are concerned with is x²+ bx + 1
= 0. An animation of this equation by varying *b* is shown below.

Figure 1: Varying *b* in the equation
y = x² + bx + 1

From figure 1 above, it shows that varying *b*
shifts the graph along the x-axis. More detailed observations shown here.

We can also view a graph of this equation along
the xb plane by solving for *b*. Solving for b gives the equation

b =( -x² -1)/x

A graph showing this equation is shown below

Figure 2: b = ( -x² -1)/x

Now lets investigate the graph when b is any number. Let b = 4

Now we see that we have two real roots at b = 4.

What happens if we change the sign of our equation from c = -1 to +1

Click here

We see that the y-axis still serves as a vertical assymptote to the graph. Now lets change the value of c and b also and investigate any differences

Click here

We see that the graph shifts but still uses the y-axis as the vertical assymptote. Furthermore, the zero's of the graph change.