Further Explorations with Parabolas

By:Lauren Lee

 

 

Letís look at y = ax2 + bx + c again.Set a = 1 and c = 1 and vary b.Letís see what happens when b = -5, -3, -1, 0, 1, 3, 5.

 

 

b = -3, b = -2, b = -1, b = 0, b = 1, b = 2, b = 3

 

 

 

We can see that all of the parabolas pass through the same point on the y-axis, (0, 1).We will also notice for b = 0, the parabola is centered around the y - axis.Also we noted before that as the value of b decreases, the parabola dips further down to the right.Similarly, as b increases, the parabola dips further down to the left.

 

Now letís explore the locus of the vertices of these parabolas that are graphed.The vertices are:

 

(1.5, -1.25) for b = -3

 

(1, 0) for b = -2

 

(0.5, 0.75) for b = -1

 

(0, 1) for b = 0

 

(-0.5, 0.75) for b = 1

 

(-1, 0) for b = 2

 

(-1.5, -1.25) for b = 3

 

 

Letís plot the vertices and see if we can make any conjectures.

 

 

 

First notice that the vertices appear to form an upside-down parabola.Also from looking at the vertices, we can see that the roots of this new parabola are x = -1 and x = 1.

 

 

Letís try to find the equation of this concave down parabola.

 

 

 

 

 

Using our original equation, y = ax2 + bx + c, we know that a must be negative since it is concave down.So in this case, a = -1.††† So to solve the equation set y equal to the roots.

 

y = (x + 1)(x Ė 1)

 

By simplifying we get y = x2 - 1.

 

But since parabola is concave down, the locus of the vertices when a = 1 and c = 1, is the parabolay = -x2 + 1.

 

 

 

 

 

 

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