Assignment 2: Quadratic Equations


Mike Cotton

In this assignment, the quadratic equation ax2 + bx + c where the constants a, b, & c are varied will be investigated.
Let's begin the investigation with the equation ax2 + bx + c where a = 1, b = 1, and c is varied. Let's look at equations where c equals: -3, -1.5, 0, 1, & 2.5.

y=1x^2 + 1x + 0

This graph was created by Graphing Calculator 3.2.

There are several patterns that are noticable. One is that as c becomes larger the corresponding parabola is further up the y-axis on the graph. Another item of notice is that the parabola for each equation intersects the y-axis at its value of c. For example the value for c in the second equation is -1.5, and its corresponding parabola crosses the y-axis at -1.5. This makes sense because x equals zero along the y-axis, so if x = 0 is used in all the equations above then y would equal c. Also notice that although all of the parabolas are at different positions on the graph, they are all the same size.

Now let's investigate the same quadratic equation where this time a = 1, c = 1, and b is varied.

This graph was created by Graphing Calculator 3.2.

Notice that every one of the parabolas pass through the point (0, 1). You can probably guess from the previous investigation that this is because c = 1 for all the equations in this investigation. Also notice that these parabolas, like the parabolas in the previous investigation, are all the same size. Another pattern that is noticable is that as b increases, the vertex of the parabola moves up and to the left until it passes through the point (0, 1) that was just mentioned. Then the vertex continues to move left but now downwards. This will be investigated in more details later in this assignment.

The next investigation of the quadratic equation will look at the case where b = 1, c = 1, and a is varied.

This graph was created by Graphing Calculator 3.2.

The first item to notice in this investigation is that all of these parabolas are of different sizes, and there is a line instead of a parabola for the third equation. Again notice that all of the parabolas pass through the same point on the y-axis. It is also worth noticing that when a is positive the parabola opens upward, and if it is negative it opens downward.

In the investigations above the form of the equation has been ax2 + bx + c. There are other forms to represent the same equation. One form is
y = a(x + (b – (b2 – 4ac)1/2)/2) (x + (b + (b2 – 4ac)1/2)/2a), where
(b ± (b2 – 4ac)1/2)/2a  is the quadratic equation. This form is helpfull when determining the points there the parabola intersects the x-axis.

Another form is
4p(y – k) = (x – h)2, which was mentioned in Assignment 1. This form is useful when investigating the parabola geometically. The variable p determines the focus of the parabola which effects the shape of the parabola, and whether the parabola will open up (positive p) or down (negative p). The variables h and k are the x & y coordinates for the vertex, (h, k), of the parabola. The vertex is the lowest point of the parabola if it opens upward, and the highest point if it opens downward. Let's take a look at some parabolas written in the form 4p(y – k) = (x – h)2.

This graph was created by Graphing Calculator 3.2.

In this investigation the values for p (0.25) and h (-0.5) are constant, while the value for k varies. Since k is the y-coordinate for the vertex of the parabola, the parabolas are all the same size (constant p), all have the same x value (h = -0.5) for the vertex, but are at different y values. As it turns out, this graph is the same as the first graph in this assignment. The variables p, h, and k can all be represented in terms of a, b, and c from the equiation ax2 + bx + c.  p = 1/4a, h = -b/2a, and
k = c – b2/4a. It can now be understood that if the variable c is changed this only effects k, which is the y-coordinate of the parabola's vertex. If the variable b is changed, this effects  h & k which are the x and y coordinates of the vertex. If the variable a is changed, this effects p, h, & k. With this explained it is suggested that the previous investigations be reviewed.

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