Varying Coefficients in a Quadratic
We are going to explore the graph of a quadratic equation y = ax²+ bx + c when a = 1 and c = 1. Thus, we have the equation y = x²+ bx + 1. On an xy plane, the graph of this equation is a parabola with a y-intercept of 1. A movie of the parabola with b values ranging from -5 to 5 is shown below.
Depending on the value of b, the parabola sometimes has two real roots, sometimes it has one real root, and sometimes it has no real roots. A root of the equation is defined as the value of x when y = 0, thus a real root of the equation is the x-value when the graph intersects the x-axis.
What can we learn when the equation is graphed on the xb plane?
Let us now consider the graph of y = x2 + bx + 1 when it is graphed on the xb plane instead of the xy plane. This graph is shown below.
Assume now that b = 4, which is a horizontal line through 4 on the b-axis. This line intersects the graph in two places, as shown below.
On the xy-plane, the graph of the parabola for y = x2 + 4x + 1 (when b = 4) crosses the x-axis in two places. We can find those two values, or roots, by using the quadratic formula.
The decimal approximations of these two roots are x = -.268 and x = -3.732. If we trace the graph on the xb-plane, we discover that the intersections of y = x2 + bx + 1 and b = 4 have the same x-values of the real roots of the equation in the xy-plane. See the graphs below.
We can then make the conjecture that the x-values of the intersection(s) of the line b = n with the equation for y = x2 + bx + 1 on the xb-plane will yield the same x-values as the real roots of the equation. We can see from the picture that when b > 2, there will be two negative real roots to the quadratic. Below is a picture of several lines when b > 2. Each of them intersects the curve in two places with negative corresponding x-values.
Let us examine what is happening in the quadratic equation when b > 2. Since a = 1 and c = 1, then the value inside the square root (which we call the discriminant) will always be: b2 – 4. Thus, if b > 2, then this value will always be positive and the square root will always be a real number. This then yields two roots because –b ± a real number will give us two real numbers (and then we would divide them by 2). Similarly, if b < -2, then the discriminant will always be positive and the equation will have two positive real roots. Below is a movie showing b varying from -5 to 5.
We also can observe from the graph that if b = 2, there will be one negative real root. In the case that b = 2, the discriminant will be 22 – 4, which is always 0. Since the square root of 0 is 0, we will get one root and it will always be –b/2. Similarly, if b = -2, there will be one positive root, x = b/2.
If -2 < b < 2, then the graphs of b = n do not cross the graph of y = x2 + bx + 1 and thus there will be no real roots. This makes sense with the quadratic equation since the discriminant is negative if -2 < b < 2 because b2 – 4 < 0. Since the square root of a negative number is imaginary, there will be no real roots.
What can we observe when the value of c is changed?
Let us change the value of c from 1 to -1 graph the resulting equation on the xb-plane.
We can see that no matter what horizontal line we draw (b = n), there will always be two real roots for the equation. This makes sense with the quadratic formula since a value of c = -1 yields a discriminant of b2 - (- 4) = b2 + 4, which will always be positive no matter the value of b. Thus, there will always be two roots to the equation. For what other values of c will the equation always yield two real roots? Since we understand the discriminant, there will always be two roots when c < 0. Let us draw several graphs on the xb-plane when c < 0.
The gray line is the line formed when c = 0. We can see that for any c ² 0, the parabola will always have two real roots no matter the value of b.
If the value of c is greater than 0, the number of roots will depend on b. Several graphs with c > 0 are shown below on the xb-plane.
Notice that for each of these graphs there are certain values of b that will yield no real roots.
What is the significance of the line 2x + b = 0?
Let us graph 2x + b = 0 on the same xb-plane as the quadratic y = x2 + bx + 1 and b = 4.
It appears that the intersection of this line 2x + b = 0 and the line b = 4 is equidistant from the two intersections of b = 4 and y = x2 + bx + 1. In other words, the x-value of the intersection of the line and the horizontal line b = 4 is equidistant from the roots of the equation when b = 4. It appears that this is true for any value of b. So, the x-value of the intersection of 2x + b = 0 and b = n is equidistant from the two roots of the parabola when b = n. Let us observe this line as we vary c in the equation.
Well, the intersection of 2x + b = 0 and b = n will always occur when 2x + n = 0 or when x = -n/2, or in other words when x = -b/2. This makes sense when we consider the parabola of the equation y = x2 + bx + c. The x-value of the vertex will always be –b/2 and it is completely independent of the value of c. Also, the line x = -b/2 is the line of symmetry for the parabola which means it is equidistant horizontally from the roots of the parabola. Thus, the line 2x + b = 0 is just the line connecting all the vertices of the parabolas of the form y = x2 + bx + c.
We have already discussed that when the discriminant is 0, then the parabola has exactly one root. The quadratic formula for all quadratics of the form y = x2 + bx + c will give us x = -b/2 when the discriminant is 0. Thus, the line formed by x = -b/2 goes through the points where the parabola has exactly one root, which is the vertex. Thus, the line x = -b/2 (or 2x – b = 0) is the line that intersects all of the vertices of the parabolas of the form y = x2 + bx + c.