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#3

## The locus of vertices of y = ax^2
+ bx + c

## for varying values of b.

*Claim:*

For fixed **a** and **c** and varying **b**,
the function **y =** **ax^2 +
bx + c** will have a locus of vertices that is also a
parabola with equation **y =** **-**
**ax^2 + c**.

*A Proof:*

Use parametric equations. That is, we
know that the locus of vertices must always have x = -**b**/2**a** and y = -**b**^2/4**a** + **c**. [Note that
calculus can be used to show that x = -**b**/2**a** is the x-coordinate of the vertex of a parabola
in the form y = **a**x^2
+ **b**x + **c**; plug in the x-value to find the corresponding
y-value, -**b**^2/4**a** + **c**.]

Now, solve both equations for **b**:

-2**a**x = **b**

+ or - sqrt(4**a**(**c** - y)) = **b**

and set them equal to each other: -2**a**x = + or -
sqrt(4**a**(**c** - y))

Squaring both sides, we have

4**a**^2x^2 = 4**a**(**c** - y)

**a**x^2
= **c**
- y

y = - **a**x^2 + **c**.

*Significance:*

The zeros of **y =** **-** **ax^2
+ c** will
be real only when - **a** and **c** are opposite in sign, i.e., when **a** and **c** are the same
sign. When **a**
and **c**
ARE either both positive or both negative, the zeros of **y
=** **-** **ax^2
+ c** are
x = sqrt(**c**/**a**) and x = -sqrt(**c**/**a**). These zeros
are the values of x that are single real roots for some specific
values of **b**
in the quadratic equation **ax^2
+ bx + c** **= 0**. That is, these zeros indicate the
x-values for the values of **b** that make the graph of **y =** **ax^2 + bx
+ c** tangent
to the x-axis.