Exploring the B
By: Russell Lawless
We will be exploring the patterns of the quadratic equation. The reason we are not focusing on the roots of the quadratic equation is because that is what most people like to find and use with this type of equation. We know that the quadratic function is
y = ax2 + bx + c
Many of us have explored this function in the xy plane, but this exploration will be for exploring in the xb plane. However, with the technology that I am using I will be substituting my b parameter for y. This will keep everything moving in a more flowing manner. So my equation will be ax2 + yx + c.
So let's focus on the graph of x2 + yx + 1 = 0. We see that we get a hyperbola.
Let's look at multiple graphs of a varying c where 0 ≤ c ≤ 4.
You can see that when c is equal to 0 that it creates an asymptote here for the hyperbolas. It appears that asymptotes are located at the y-axis and x = -y. It seems when c is positive that the graph will be on the top and bottom the the X that is formed from the two asymptotes.
Now let's look at multiple graphs of a varying c where -4 ≤ c ≤ 0.
Again we see that when c is equal to 0 that the asymptote is formed. It appears that these sets of graphs have the same asymptotes as before. It seems that when c is negative it goes on the left and right side of X that is formed by the two asymptotes.
Put it all together!
In all cases, except when c is 0, a hyperbola is formed. When c is 0, we have a linear equation, x = -y, that is formed.
Here we want to look at what happens at different values of y (remember that y represents b in our quadratic equation).
For interpretation of this graph wherever there is a parallel line to the x-axis (all new equations), if it intersects the curve in the xy (xb) plane then the intersection of these points will correspond to the solutions of the original equation for the specific value of y (which is b). So if we are looking at all possible values of y (b), we want to figure out the "pattern" of the amount of solutions depending on the value of y. For this last part we will change our y back to a b so that we understand the final conclusion. So we can see that when -2 < b < 2 there are no real solutions, when b = -2 (will be a positive solution) or b = 2 (will be a negative solution) there is one real solution, and when b > 2 (will be negative solutions) or b < -2 (will be positive solutions) there are two real solutions.