** Write-Up #2**:

This write-up investigates variations of the quadratic equation of

__Part 1: Coefficient a__

Although coefficient a can take on all real
values from minus infinity to plus infinity, let us observe some
positive values of **5, 2, -2 **when b = 1 and c = 1. The equations
are

The first equation is graphed in purple, the second in red, and the third in blue.

There several conclusions that can be made:

(1) Since the coefficient is positive, the "arms" of the parabola open upward.

(2) All of the curves have a

y-intercept of one.(3) The

largestdistance between arms (third equation, graphed inblue) has acoefficient of .2, as compared to thesmallestdistance between arms (first equation, graphed inpurple) has acoefficient of 5. Hence, theto distance between arms.coefficient is inversely related

If the **coefficient a** of the squared
term is ** negative**, such as -5 and -2, then the graphs
will open

Lastly, if the **a-coefficient**, is 1 (the
**black** line) or zero (the **yellow** line), then the graphs
will still pass through point (0,1). The black line will appear
as a "standard" (proportioned) parabola. The yellow
line (with coefficient of zero) no longer has a squared term;
hence, it merely represents a line.

__Part 2. Coefficient b__

Coefficient b can also assume all real values.
We will observe b-values of **5 (in purple), 3 (in red), 0 (in
blue), -3 (in green), and 5 (in light blue)**. The positive
values shift the curves in the opposite direction (left), and
the negative values cause a shift to the right. All curves pass
through point (0,1).

__Part 3: Constant c__

As a final investigation, we can now investigate the c-coefficient. This value can assume all real values, but we will investigate values of -3 < x < 3:

It can be noted that the **c-variable**
corresponds to the y-intercept. For example, when c= -3, the y-intercept
is at -3 (as illustrated by the **purple** curve), and when
c= 1, the y-intercept is at 1 (as illustrated by the **light-blue**
curve).

__Part 4: Conclusion__

The a, b, and c values have very specific purposes.
The **a-value** serves to provide a scaling constant in an
inverse relationship with the arms of the parabola and to open
the arms up or down. The **b-value** serves to shift in the
opposite direction of the sign of the coefficient. Finally, the
**c-value** serves as the y-intercept in the same direction
as the sign.