## Some Different Ways to Examine

### by James W. Wilson and Melissa Silverman University of Georgia

It has now become a rather standard exercise, with availble technology, to construct graphs to consider the equation

and to overlay several graphs of

for different values of a, b, or c as the other two are held constant. From these graphs discussion of the patterns for the roots (or where the graph touches the x-axis) of

can be followed. For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation). The graph's behavior on the x-axis has three different trends, which translate into zero, one, or two roots for the equation.

1. For b < -2 the parabola will intersect the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive). For b > 2, the parabola intersets the x-axis twice to show two negative real roots for each b.

2. For b = -2, the parabola is tangent to the x-axis and so the original equation has one real and positive root at the point of tangency. Similarly for b = 2 the parabola is tangent to the x-axis (one real negative root) and

3. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots.

There will never be more than two roots because the equation's highest term is squared. The number of roots an equation has is less than or equal to the degree to which it is raised. A squared equation is a parabola, which can only cross the x-axis a maximum of 2 times due to its shape.

Now consider the locus of the vertices of the set of parabolas graphed from

.

Following the vertices of the set of parabolas as the value of b changes suggests that the locus is a parabola that opens downward. The locus is the parabola

Where the 1 indicates the one point on the y-axis that the set of parabolas share. The x term is negative because the parabola faces downwards.

In general, the formula for the locus of the vertices of a set of parabolas will have a c term that is equal to the common y-value that all of the parabolas share. The orientation of the parabola (and thus the sign of the x term) will be the opposite of the set of parabolas' orientation. Below is another example of a set of parabolas. Here the c term is now 2, and -2<b<2. The locus of the vertices (the yellow parabola below) is the parabola

### Graphs in the xb plane

Consider again the equation

Now graph this relation in the xb plane. We get the following graph.

If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b.

For each value of b we select, we get a horizontal line. Similar to our previous results, the line can intersect the curve zero, one, or two times, meaning that there can be zero, one or two roots.

1. We get two negative real roots of the original equation when b > 2 and two positive real roots when b < -2, meaning that the horizontal line cuts through one of the curves twice.

2. We get one negative real root when b = 2 and one positive real root when b = -2, meaning that the horizontal line is tangent to the top of one of the curves.

3. We get no real roots for -2 < b < 2, meaning that the horizontal line never touches the curve.

Consider the case when c = - 1 rather than + 1, where the red curves represent the graph and the green curves are asymptotes.

In this scenario, there will always be two roots. The blue horizontal line (which represents the value of b) can move up and down the y-axis and always touch the red curve in two places. There is no value of b where the horizontal line will be tangent to (with one root) or not touching (with no roots) the red curve. But why?

Let's consider what it means for a graph to have a c value of -1. This means that the vertex of the graph has a negative y value. The vertex is below the x-axis. The leading coefficient, x^2, is positive, meaning that the parabola faces upwards. If the vertex of the parabola is below the x-axis but the parabola faces upwards, the sides of the parabola will have to cross the x-axis, thus there will always be two real roots.

For example, consider the equation

When the vertex is below the x-axis, but opens upwards, you can't avoid crossing the x-axis twice! Thus you will always have two real roots.

This brings up an important relationship between the orientation of the parabola and the placement of the vertex. One of five things may occur with a parabola.

1. If the parabola faces upwards and its vertex is above the x-axis, then there are no real roots.

2. If the parabola faces downwards and its vertex is above the x-axis, then there will always be two real roots.

3. If the parabola faces upwards and its vertex is below the x-axis, then there will always be two real roots.

4. If the parabola faces downwards and its vertex is below the x-axis, then there are no real roots.

5. If the vertex lies on the x-axis, then there is only one real root, no matter the orientation of the parabola.

Graphs in the xc plane

In the following example the equation

is considered. If the equation is graphed in the xc plane, it is easy to see that the curve will be a parabola. For each value of c considered, its graph will be a line crossing the parabola in 0, 1, or 2 points -- the intersections being at the roots of the orignal equation at that value of c. In the graph, the graph of c = 1 is shown. The equation

will have two negative roots -- approximately -0.2 and -4.8.

Just as in previous examples, we can see that depending upon the value of c, there will be zero, one, or two roots.

1. For c > 6.25, the equation will have no real roots.

2.There is one value of c where the equation will have only 1 real root -- at c = 6.25, the vertex of the parabola.

3. For c < 6.25 the equation will have two roots. The roots are negative for 0 < c < 6.25, one negative and one 0 when c = 0, and one negative and one positive when c < 0.

The c value determines the y intercept of the graph. When the graph faces upwards (like the one in this example), and c is negative, then the vertex is below the x-axis. This means that there are two real roots. There are also some positive values of c for which there are two roots, meaning that the y intercept is positive but the vertex is still below the x-axis. When the value of c is 6.25, the vertex of the graph is directly on the x-axis, so there is only one root. Any value of c that is higher than 6.25 means that the vertex of the graph is above the x-axis. As stated previously, a parabola that faces upwards with a vertex above the x-axis has no real roots.

Graphing on the xb and ac plane brings one down a different path of analysis of a parabola. The more traditional system of plotting parabolas and comparing their differences helps one to begin assigning roles to each term of the equation. Investigation 3 emphasizes different characteristics, especially how the values of b and c affect the number of roots that the parabola has. I think each exploration is important, but each is certainly supported by the other. Using both approaches together to understanding parabolas provides a truly strong foundation

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