Investigation of Quadratics in the xb plane

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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

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