**EMAT 6680**

**Write-up 2**

**By
Laura King**

**This
assignment is an exploration of the parabolic equation given below.**

In this assignment, we will look at how the graph of the equation changes as two of the values of a, b, and c are fixed.The third value will be varied and we will discuss how that affects the graph.

**Varied
values for a:**

In the first set of graphs, a is varied from -3 to 3 and b and c are fixed at 1.

**Graph of**

**where a varies from -3 to 3.**

As the value of **a **changes
from **1** to **3**, the **parabolas **shape becomes more narrow
and is shifted up and to the right on the coordinate system.All
of the graphs where **a** is positive open upward on the coordinate
system.

The graph of the equation where
**a=0**
is a **line**.Because a equals
0 the equation becomes y=x+1.This
equation is a linear equation.Therefore,
the graph is a line that intersects the y-axis at the point (0,1).This
point is called the **y-intercept** and can be found using the slope-intercept
form of a line **y=mx+b**, where b is the y-intercept.

As the value of

You should also compare the **parabolas**
of the values of **a** and their opposites.Ifyou
compare the graph of the equation where **a=1** and **a=-1**, you
will see they have the same shape but open in opposite directions on the
coordinate system.The same is true
for the graphs where **a **equals **2** and -**2** and the graphs
where **a **equals **3 **and -**3**.

Therefore, we can tell from these
graphs that **a** does effect the shape of the **parabola**.It
also shifts the graph slightly to the left and right and up and down along
the coordinate system.**A **also
effects whether the **parabola** opens up or down.When
**a**
is positive, the graph opens upward, and when **a** is negative, the
graph opens downward.

In the next set of graphs, **b **is varied from -**3** to
**3**
and **a** and **c** are fixed at **1**.

**Graph of**

**where b varies from -3 to 3.**

As **b **varies from -**3 **to
**3**,
the **parabola** is shifted up and down and to the right and left on
the coordinate system, but the shape of the
**parabola** stays constant.

**Key:**

**b=1 green
b= -1 red b=0 blue**

**b=2 light blue
b= -2 purple**

**b=3 yellow
b= -3 pink**

As **b** changes from **1 **to
**3**,
the **parabola** is shifted to the left of the y-axis and further down
on the coordinate system.As
**b
**changes
from -**1** to -**3**, the **parabola** is shifted to the right
side of the **y-axis **and also further down on the coordinate system.The
graph where b=0 is centered along the **y-axis**.All
of the
**parabolas** open upward because **a **is positive.Also,
all of the **parabolas** pass through the point (0,1); just like the
graph in the first set where **a **was varied.

You can also compare the values
of **b **and their opposites.The
graph where **b=1** and **b= -1** have the same **y **values but
are on opposite sides of the **y-axis**.**B=1**
is on the left and **b=-1** is on the right.The
same is true for the graphs of **b=2** and **b= -2** and for **b=3**
and **b= -3**.

Therefore,
we can tell from the graphs that the value of **b** does not change
the shape of the **parabola**.It
does affect the shifting of the **parabola** up and down and left and
right on the coordinate system.**B**
also does not effect whether the **parabola** opens up or down.

**Varied
values for c:**

In the last set of graphs, **c** is varied from -**3** to
**3**
and **a **and **b** are fixed at **1**.

**Graph of**

**where c varies from -3 to 3.**

As **c **varies from -**3**
to **3**, the shape of the **parabola** stays the same but is shifted
up and down along the **y-axis**.

**Key:**

**c=1 green
c= -1 red c=0 blue**

**c=2 light blue
c= -2 purple**

**c=3 yellow
c= -3 pink**

** ** As **c** varies
from **1 **to **3**, the **parabola **is shifted further up along
the **y-axis**.As **c **varies
from -**1** to -**3**, the **parabola **is shifted further down
along the **y-axis**.The parabola
where **c=0 **is centered close to the **origin**.As
stated earlier, each parabola has the same shape when c is changed from
-**3** to **3**.Also, the
graph is not moved to the left or right as **c** is changed from
**3**
to -**3**.

Therefore, we can see from this
graph that **c** effects the movement of the graph up or down along
the **y-axis**.**C** does
not effect the shape of the graph nor the movement of the graph to the
right or the left.

**In conclusion, we have seen that in the parabolic equation **

**each variable a, b, and c performs a different function in the graph.**