Some Different Ways to Examine

ax2 + bx + c = 0

 

By

 

James W. Wilson and Audrey V. Simmons

University of Georgia

 

 

It has now become a rather standard exercise, with available technology, to construct graphs to consider

the equation ax2 + bx + c = 0.  We will be looking at some of the relationships that exit in the world of parabolas.

 

 

PART I

 

Our starting point becomes y = ax2 + bx + c 

 

For a = 1, c = 1  our equation is  y = x2 + bx + c.   We will vary b. 

Let  -3 b 3

 

y= x2 +1    y= x2 + 1x + 1  y= x2 +2x +1   y= x2 +3x +1  y= x2 -1x +1

 

 y= x2 –2x + 1   (Yellow)      y= x2 –3x + 1  (gray)

 

 

 

 

 

 

We can see that all of the graphs have a y-intercept of (0,1).  Would we have a y- intercept of 3 if c=3 and the equation was y=x2 + bx +3 ?

 

Click HERE to see if there is a common y-intercept.

 

Would the same be true when a=2 and c=3 for y= 2x2 +bx + 3?

Click HERE to see.

 

Surprise!  No matter what the coefficients of a or b, the value of  c  tells us what the y-intercept is.  There is a movement of the parabolas through the same point on the y-axis.

 

Please look back at the original graphs above.  At the points where b=2 and b= - 2 the graphs are tangent to the x- axis. Recall that the roots of the equation indicate where the graph touches or crosses the x -axis.

The roots of the equations are as follows:

                  y = x2 +1            no roots                         

 

                  y = x2 + x + 1    no roots                  y = x2  -1x + 1  no roots

 

                  y = x2 + 2x + 1   one root                          y = x2 -2 x + 1   one root

 

                  y = x2 + 3x + 1   two roots                         y = x2 -3 x + 1   two roots

 

When b> 2, there are two negative real roots.  When b<-2, there are two positive real roots.  When –2< b < 2, there are no real roots. 

 

 

PART II

 

Is the locus of the vertices of a set of parabolas a parabola?

Look at the black graph of  y = -x2 +1.  It appears to travel through the vertices of each parabola.

 

 

If we found the vertices of each of the graphs, they would be solutions for the equation y = -x2 +1

 

PART III

 

Consider the same equation x2 +bx +1 = 0.  We will graph this equation in the xb plane.  That means we will solve for b instead of y giving us the equation

  b =     or       b  =

 

x will still be found on the horizontal axis and b will be found on the “y” axis.

 

What are the roots of the graph?  Our quadratic equation x2 +bx +1 = 0 graphs as a hyperbola. When b = 2 or b = -2 we have one root at the vertex of the hyperbola.  When b = 3 we have 2 negative roots. Therefore, for all values of b > 2, there are 2 negative roots. When b < -2, there will be 2 positive roots.

 

 

When c =-1, values less than –1 approach a diagonal asymptote and the vertical axis.

 

 

 

 

PART IV

This time we will graph our equation in the xc plane.  Our equation is

 

x2 + 5x +c = 0         or           c = -x2 – 5x

 

This will be the graph of a parabola.  In the graph below x is represented on the horizontal axis and c is represented on the vertical axis.

 

 

x2 + 5x +c = 0  is our parabola   

 

c = 6.25  shows one root       

 

  c> 6.25 shows no roots

 

c=1 shows two negative roots

      

  c = 0 shows one negative root and one root of zero

 

c = -2 shows one positive and one negative root

 

To summarize when c < 0  there are two roots, one is positive and one is negative.  When c = 0, one root is negative and one is zero.  When 0< c < 6.25, there are two negative roots.  There is only one root when c = 6.25.

 

 

 

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