__Write-Up #1__:
Explorations With Graphing Relations and Functions

__Part 1: A Cursory Analysis of Variations
of the Following Equation:__

Given the following equations,

the graphs are superimposed on the same graph where the first equation is the red (burnt orange) line, the second equation is the purple line, and the third equation is the blue line.

Essentially the lines are nested for various constants following the x-squared term with the exception of the constant one (1).

__Part 2: Exploration of the General Sine
Trigonometric Equation__

We will investigate the following trigonometric equation

for the effects of different values of a, b, and c.

__Investigation of the a-Value__:

This value is the scalar constant of the trigonometric function which affects the amplitude of the function. Whatever value the sine function assumes, there is a multiplication by value a. The amplitude, which is the maximum and minimum attained values, will be the absolute value of a. If we observe the standard sine function where a=1, then the amplitude is also one. If the amplitude is changed, for example where a= 3 or 5, then the maximum and minimum values will be changed accordingly. Observe the graphs of three equations where a=1, 3, 5.

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

As is noted, the period is unaffected.

If a is positive, then the sine function, which
has a zero value for a zero x-value and immediately assumes __positive__
y-values within the first quarter of its period. If a is negative,
then the sine function will immediately assume __negative__
y-values within the first quarter of its period. These relationships
can be graphically shown by the following equation where the **blue** line
has an **a-value** **of +5**, and the **purple** line has an **a-value of -5**.

Again the period is unaffected.

__Investigation of the b-Value__:

This value is related to the period (T) by
the equation of T = fundamental period / b where the fundamental
period refers to the period of the trigonometric function being
used (sine). Hence, there is an __inverse relationship of b to
the period__. This relationship can be graphically shown by
the following equations:

If we assume that c=o, and the standard graph
is in red , the first equation has a **b-value** of **5**
as shown in **green**. The period of this equation is now **1/5** the
normal period of the sine curve. The second equation with a **b-value**
of **1/5** , the curve shown in **light-blue**, now has a period
of **5-times** the normal period.

__Investigation of the c-Value__:

The c-values affect the starting point of the
trigonometric function called the phase angle. The **phase angle
**can be found by the equation **x=-c/b**. If we begin with
an equation of **y = a sin (ax + c)**, we can set a = 5 and
b = 2, then we can investigate the different c-values.

For an investigation of different c-values, observe the following equations:

The first equation is in **blue** and has been shifted
to the **left** from a starting value of x = 0 by -c/b, whereas
the second equation is in **purple** and has been shifted to the **right** from a starting
value of x = 0 by -(-c/b) = (c/b):

Hence the values of a, b, and c have very different impacts upon the graphs as has been illustrated, above.

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