This exploration will examine the effects the quadratic equation:

We will explore the effects of each parameter by fixing the other two and noticing the effects.

For the first case we will see what the parameter a does while fixing b = c = 0.

If a = 1 then y = x². This corresponds to the below graph:

The next graph corresponds to the overlapping of a = 2 & a = 1while b = c = 0.

We can see that an increase in a leads to an increase in the "steepness" or slope of the parabola.

The next graph corresponds to a = 1, 2, 3, 5:

Letís try smaller values and see what happens. Say a = 1, .5, .3, .1, .01 (from inside to outside respectively)

It is as we thought. The parabola is getting wider for a smaller and more narrow for a larger.

What about a negative?

Usually negative signs cause some form of reflection in Euclidean space.

Letís think about this, if y = -(some #) x², then x² is always positive, and then we are taking the negation of this.

So for each x value both positive and negative, are y value will be a negative number (except of course for x = 0).

So the parabola will be flipped over the x-axis.

Wow, our thinking about this was right. Now what do you think would happen if a were a large negative number as opposed to if -1 < a < 0?

Well the only difference that the negative sign made on y = x² was to flip it over the x-axis.

It may be safe to predict that a large negative number for a will increase the steepness of the slope, and a fractional value for a, will widen the parabola or make the slope smaller.

Lets check,

This was for b = c = 0 lets. The reason we chose b = c = 0 is because x² is the highest degree term in our equation

and will dominate all the other terms for large x, that is, this term will "govern" what the function looks like as x

goes to infinity & a is itís coefficient.

Let us explore some graphs with other possible variances of fixing b and c while changing a.

We can easily notice the straight line on this graph. This line is when the value of a = 0. This is a linear equation in

the form y = mx + b (in this case y = x + 1).

Before we investigate what b and c do let us look at one more overlapping graph:

This investigation will only cover cases where a, b, and c are constants, but exploring some cases where the coefficients are functions can be of great importance! Notice that the picture still has the general resemblance of a parabola. This is because of the leading degree (power) as discussed before. Let us now look at what the value of b does to our graphs.

Let us first examine the case of a = c = 0 and b = 1 or y = x.

Is this a good graph to investigate?

Not really, because when a = c = 0 we just have a straight line which does not really tell us how the overall function will behave.

Let us start with a something more interesting like a = c = 1 and vary b.

Say b = 1 and 2 separately and then overlapped.

For now let us keep the parameter c = 0 and look at different cases of y = x² +bx (for a =1 b = 0, 1, 2, 3, 4, 5 c = 0).

This is because I want to see how just the addition (or subtraction) of bx onto x² affects x².

Wow! We see the y = x² equation, and then for each successive b value increased the parabola has descended below the x-axis so that itís ëleft-mostí x-intercept corresponds to the negative b parameter, where as the right most intercept has remained fixed at the origin. B seems to be moving the vertex of our parabola.

Letís think about this:

    1) We are looking at the graphs of quadratic equations.
 
    2) When I think of quadratic equations I think of the quadratic formula:

x = [-b±sqrt(b²-4ac)] / 2a

This will give us the roots for x, that is, where the parabola will intercept the x-axis.

Now this will happen either once, twice or not at all. The discriminant will tell us this.

The discriminant is d = b² ó 4ac:

    If d is negative the parabolaís vertex lies above the x-axis and there are two distinct conjugate complex roots (but

    no real roots).

    If d is positive the parabolaís vertex lies below the x-axis and there are two real roots (call these zero's x₁, x₂ ).

    If d is zero then the parabolas vertex lies on the x-axis and there is one real zero (a double root).

    Now back to finding our vertex. It is easy to see that when d=0 our x roots correspond to -b/2a.

    Which gives us a double root for x.

    If f(x) = ax² + bx + c and (b² - 4ac = 0) then the vertex lies at the point (-b /2a, 0).

    If f(x) = ax² + bx + c and (b² ó 4ac > 0) then the vertex lies below the x axis and has x intercepts of x₁, x₂

    which correspond to real roots roots of the quadratic equation and our vertex lies at [(x₁ +x₂)/2, f((x₁ +x₂)/2)].

    If f(x) = ax² + bx + c and (b² ó 4ac < 0) then the vertex lies above the x-axis and has no x-intercepts!

    At y define the horizontal line y = x₁. Now follow y to itís other intersection of the parabola (we can do this by

    plugging in y into our equation and solving for x) call this other intersection x₂.

    Now we have two distinct points on our parabola and the vertex can be found as V = [(x₁ +x₂)/2, f((x₁ +x₂)/2)].

    X-ploration 3 covers more on this.

So now we know what the a parameter does (it modifies the slope) and what the b parameter does (it moves the vertex). We also have another term called the discriminant which we have used to help us. Now we need to find out what c does. Letís look at the following graph:

This makes sense because if all the "x's" were to be zero, then our equation would just be y = some number be it positive or negative or zero. We are just adding on a scalar quantity to y. Notice again how all of our parabolas have a similar shape.

Definition: Scalar ó Usually, a real number, particularly in operations with vectors.

Definition: Parabola ó A conic with eccentricity 1; it is defined as the locus of points P whose distance from a fixed point (the vertex) is equal to itís distance from a fixed line (the directrix). The parabola is symmetrical about it ís axis, which is the line through the vertex and perpendicular to the directrix; the point of intersection of the axis with the curve is the vertex V.