Assignment Three

The Joy of Parabolas

Collaborative Effort by

Lucia Zapata and Claudette Tucker

We will explore the quadratic formula of when b is different values.


Let's examine this equation when b = -3, -2, -1, 0, 1, 2, 3.


What can you infer from the graph above?

The graph always passes through the same point, (0, 2) on the y-axis.

When b < -2, the parabola will have two positive real roots.

When b = - 2 and b = 2, the parabola's tangent is the x-axis.

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

How many real roots are there a b = -2 and b = 2? Are the roots positive or negative?

What happens when b > 2?

Do you think that the vertices of these parabolas share something in common?

Yes, they have something in common. We will prove this by showing that the locus is the parabola .

Before we examine their relationship, we should define locus. Click here for definition

Let's use this information to explore the relationship of the parabolas.

 

Let's see if we can pick some points to verify the locus.

We will the use the points (0, 1), (-1, 0), and (1,0) to form a system of equations using quadratic formula,

This implies that c = 1.

This implies that a - b + c = 0.

This implies that a + b + c = 0.


We will use c = 1 and the sum of the second and third equations to solve our system of equations.

a - b + c + a + b +c = 0 This implies that 2a + 2c = 0. Since we know that c = 1, we can solve for a. a = -1. Let's use the facts to determine the value of b. -1 + b +1 = 0 This implies that b = 0. Thus, is the equation that passes through the points (0, 1), (-1, 0), and (1,0).

Let's examine the graph of and .

Therefore the locus of the vertices of the set of parabolas form a parabola as well at .

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