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 a varies from -3 to 3, the parabola changes shape and is shifted on the coordinate system.

Key:
a=1 green            a= -1 red    a=0 blue
a=2 light blue      a= -2 purple
a=3 yellow          a= -3 pink
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 a changes from -1 to -3, the parabolas shape also becomes more narrow and this time is moved down and to the left on the coordinate system.All of the graphs where a is negative open downward on the coordinate system.
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.
Varied values for b:

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.