Assignment 3

This activity explores of the equation ax² + bx + c = 0, and the effect different values of a, b, and c on its roots.

Let y = ax² + x + 1.  The following picture shows the graphs of this function when  a = -3, -2, -1, 0, 1, 2, 3.

In all cases, the graph intersects the y axis at y = 1.  When a > 0, this function has no real roots.  When a < 0, however, the function has two real roots.  If we imagine the parabola starting at a = 0, then as the value of a increases, the parabola gets narrower and narrower in an upward opening direction.  Since these parabolas will never cross the x-axis, they have no real roots.  If we again imagine starting at a = 0, or a straight line, then as the value of a decreases, the parabola becomes narrower and narrower in a downward facing direction.  These parabolas will always cross the x axis, and will therefore always have two real roots.

Now, let y = x² + bx + 1.  The following picture shows the graphs of this function when  b = -3, -2, -1, 0, 1, 2, 3.

Again, these graphs all intersect the y axis at y = 1. Changing the value of b seems to have two impacts. 1) As the value of b falls further and further from the x axis, the minimum of the function becomes increasingly smaller.  2)  As b becomes increasingly more positive,  the minimum of y becomes increasingly negative, relative to the x axis.  As b becomes increasingly negative, the minimum of y becomes increasingly more positive, relative to the y axis.  When |b| = 2, the function will have one real root.  When b < -2, the function will have two real negative roots.  When b > 2, the function will have two real positive roots.  When -2 < b < 2, the function will have no real roots.

Finally, let's look at the function y = x² - x + c.
The following picture shows the graphs of this function when  c = -3, -2, -1, 0, 1, 2, 3.

In this graph, notice that every parabola is symmetric about the line x = 1.  In fact, the set of all vertices of this parabola, where c is a real number, would be the line x = 1.  When c = 0 or c > 0, this function has two real roots, one negative and one positive.  When c < 0, this function has no real roots.

Now consider the locus of the vertices of the set of parabolas graphed from
y = x² + bx + 1.  Notice that the graph of this set of vertices is the parabola y = -x² + 1 (shown in light gray).

Let's look at one more case like this.  Consider the function y = -x² + bx - 5, where b = -3, -2, -1, 0, 1, 2, 3.

Notice this time that the graph of this set of vertices is the parabola y = x² - 5 (shown in dark gray).

We can generalize then, than the set of the vertices of all parabolas of the form y = x² + bx + c, is the parabola, y = -x² + c.