Sarah Hofmann
EMAT 6680 Summer 2006
Assignment 3
Investigation 1

Our assignment for investigation 1 was to notice that all equations of the form Parabola with constant coefficient = square coefficient = 1 and linear coefficient = bhave their verticies on the equation (parabola with constant coefficient 1, square coefficient -1, and linear coeficient 0)as shown in the graph below.

Graph showing the above concept.

We now need to generalize this idea. 
Investigation 2

We were to graph the equation general parabolawhere a = 1 in the xb plane for different values of c.  We have chosen to plot the integers c = -4 . . . 3

Graphs stated above.

Notice that in the case where c = 0, the turquoise line, is linear. 

Investigation 3

Our job for this investigation is to add the equation Equation 3to the above graph and investigate its relationship to the quadratic formula.  We will add the equation to the above graph in the form Equation 2..  It will appear as a purple line.

 Graphs stated above.

The quadratic formula for our equation with a = 1 is . If we look at the quadratic equation in the context of the line , we can see the following

But we know that implies that the equation has only one root.
Investigation 4.

We are to consider graphs in the xc plane for this investigation.

If we graph several basic parabolas in the form  for integer values of c=-4...3, we can see the following graph.

If we differentiate the basic equation, solve for y', and set y' = 0, we can see the first derivative is zero when y = x^2.  So the local maxima occur on this parabola.  Indeed if we add the equation y=x^2to the above graph we see it intersects all the parabolas at their verticies.  The new parabola is in purple.

Now if we consider graphs of the form ax^2 + x + c = 0for values of a = -4...3 we can see the following graphs.

Differentiating the equation ax^2+x+y=0, solving for y', and setting y'=0 we see that the critical points all coincide on the line y=-0.5x.  Indeed if we add this line as a purple line and zoom in to the above graph we can see it does cross all the local maxima or minima of the above parabolas.

Investigation 5.

We are to consider graphs in the xa plane.

If we graph several graphs of the basic equation yx^2+x+c=0for values of c=-4...3 we get the following graphs.

If we consider the basic equation yx^2+b+1=0for integer values of b=-4...3, we get the following graphs.

Differentiating the basic equation yx^2+b+1=0, solving for y', setting y'=0, and solving, we see that all the critical points lie on the equation y=1/(x^2).  Adding this equation as a purple graph to the above graph we see that indeed, all the critical points pass through the new purple line.