Exploration 3: Roots of Quadratic Functions
by Elizabeth Gieseking
All high school students know the quadratic formula and some can even sing it. They can use it to easily calculate the roots of a quadratic equation, but few know where it came from or what it really means. We will start with a derivation of the quadratic equation and then we will examine the effects of changing each of the coefficients by looking at this equation in the x-b, x-c, and x-a planes.
The most common derivation of the quadratic equation is obtained by completing the square.
Divide through by a
Complete the square
Rearrange and simplify
Take the square root of both sides
In this exploration, we will examine each of the coefficients in the standard form of the quadratic equation: and their effects on the roots of the equation. We will start by letting and we get the equation . Next we will graph this equation on the x-b plane.
We inserted several horizontal lines corresponding to different values of b. The points at which each horizontal line crosses the graph of correspond to a solution of the equation for that value of b. For example when b = 2, the line crosses the graph of the function at one point (x, b) = (-1, 2). This corresponds to the solution of which is . When b = 3, the line crosses the graph of at two points. We can use the quadratic formula to determine that these two roots are . When b = 1, the line does not cross the graph of the function . If we use the quadratic formula, we see that the discriminant which means there are no real roots.
We can also vary the value of c on this graph. Below are the curves we obtain when c is set at integer values between 1 and 6.
We see that as c increases, the lowest value of |b| which produces real roots also increases. We will again examine the discriminant to see why this is true. In order to have real roots, or . Thus, when we set a = 1 and increase c, b increases in proportion with . If we instead chose values of , then will always be greater than 4ac and we always have real roots as shown below.
In the graph above, there is one curve that does not look like the others, the curve of This curve is actually a pair of straight lines. If then either or . Plotting this on the x-b plane, we get two lines through the origin – one vertical and the other having a slope of -1. These two lines are the asymptotes of the family of hyperbolas seen in the graph.
In our final graph of b versus x, we include values of c from -5 to 5 and we also add the line
This line goes through four points which we know are single roots of our quadratic equations: (x, b) = (-1, 2) and (1, -2) on the curves corresponding to and (x, b) = (-2, 4) and (2, -4) on the curves corresponding to If we look at the horizontal b lines, it appears that the line is halfway between the two roots for that value of b. Why would this be true? If we solve this equation for x, we see that . If we look back at the quadratic formula, we see . For the equations we are examining, we are keeping a constant at 1, so this equation becomes . The two roots calculated by the quadratic formula are symmetric around .
We will look at this by graphing four quadratic equations with and
When b = 6, the roots are symmetric about . The equation has one real root at . The equation has two real roots at or -4 and -2. The equation has two real roots at or -5and -1. The equation has two real roots at or -6 and 0.
In summary, we have found that a quadratic equation of the form can have zero, one, or two real roots, depending on the value of c. If then there are two real roots symmetric about . If , then there is one root of multiplicity 2 at . If , then there are no real roots. There are however two complex roots symmetric about .
Next we will examine the equation in the x-c plane. In the following diagram, b = 4.
The horizontal lines correspond to different values of c. The points at which the graph crosses each value of c are the roots of . For example, in this graph we see that the roots of are and , has one root at , and if , there are no real roots.
We can graph the equation with multiple values of b on the same set of x-c axes.
Here we have a series of downward facing parabolas. As the value of b increases from 0, the vertex of the parabola moves upward and to the left. As it decreases from 0, the vertex moves upward and to the right. If we look back at the equation we see that the vertex is at . We also note that all of these parabolas pass through (x, c) = (0, 0). This is due to the fact that when c = 0, the quadratic can be factored as which has a root at and a second root at .
Our final exploration will take place on the x-a plane. We will start with the equation and vary b from 0 to 5.
The first thing we notice is that there is an asymptote at x =0. When x = 0, there is no value for the coefficient a which produces a solution of this equation. We see that when b > 0 and a = 0, there is one root corresponding to the linear equation If b = 0 there is no root at a = 0 since that would imply that 1 = 0. When a > 0, there may be zero, one, or two real roots, all at negative values of x. When a < 0, there are always two real roots, one positive and one negative. If we consider negative values of b, our graph is the reflection of this graph over the a-axis.
We will also consider the graph of when we keep b constant at -4 and vary c from 0 to 5.
We still have an asymptote at x = 0. When c = 0, a tends toward negative infinity when x approaches zero from the negative side and positive infinity when x approaches zero from the positive side. For all other positive values of c, a tends toward negative infinity when x approaches zero from either direction. When we choose negative values for c, the graph is reflected over the origin.