Summer Tuggle

Assignment
3

Explore
y= ax^{2 }+ bx + c

when
a and c are constant and b changes

Quadratics are polynomials of
the second degree. The word “quadratic”
seems to imply a connection to the number four.
This is not so. *Quad* refers to the classic math problem
of trying to find a square with the same area as a given circle. This is called finding the quadrature of a circle.
Quadratic, therefore, refers to finding the area (x^{2}) of a
square with a side length of x.

The standard form of a quadratic is y= ax^{2 }+ bx + c. The graph of
a quadratic is a curve that is referred to as a parabola. Let’s explore what happens when a and c are constant and b is changed.

Below is the graph of parabolas that have different “b”
values. Specifically b= 0, 1, 2, 3, -1, -2, -3

All of the parabolas intersect the y-axis at 1. When b is less than 0 the parabola’s vertex
is in the 1^{st} or 4^{th} quadrant of the coordinate plane
(the aqua, yellow and gray curves). When
b is greater than 0 the vertex falls in the 2^{nd} or 3^{rd}
quadrant (red, blue, and green curves).
The purple curve, whose vertex falls directly on the y-axis, has a b of
0. It is interesting to note that all
the vertices fall on a specific curve (the black curve).

Since we know that the x coordinate of a parabola can be
found using (-b/2a) and we know that in this case a=1 we can solve for b

X=_{}

2x = -b

-2x = b

Now take this a plug it into our standard formula where a
and c are 1, but b is constant y=x^{2 }+ bx +
1

y=x^{2 }+ (-2x)x + 1

y= x^{2} -2x^{2} + 1

y=-x^{2} +1

Therefore all the vertices will fall on the curve y=-x^{2}
+1.