
Michelle E. Chung 
*
EMAT6680 Assignment 2: Parabolas 

1. Construct graphs for the parabola
for different values of a, b, and c. (a, b, c can be any rational numbers.)


Graph of



As
we know, this is the simplest parabola graph, which is .
 The parabola
opens up.
 The vertex
of the parabola is (0,0).
 The axis
of symmetry is x=0.

When
a=b=c=1
(a,b,c>0) 

When
a=b=c=1,

The parabola opens up.
 The
vertex of the parabola is (1/2, 3/4).
 The
axis of symmetry is x= 1/2.

When a=c=1, b=3
(a,b,c>0)


When
a=c=1 and b=3,

The parabola opens up.
 The
vertex of the parabola is (3/2, 5/4).
 The
axis of symmetry is x= 3/2.

When a=c=1, b=3
(a,c>0 & b<0) 

When
a=c=1 and b=3,
 The parabola opens up.
 The
vertex of the parabola is (3/2, 5/4).
 The
axis of symmetry is x= 3/2.

When a=1, b=5, c=4
(a,b>0 & c<0) 

When
a=1, b=3 and c=4,
 The parabola opens up.
 The
vertex of the parabola is (5/2, 41/4).
 The
axis of symmetry is x= 5/2.

When
a=0.3, b=5, c=5.3
(a<0 & b,c<0) 


When
a=0.3, b=5 and c=5.3,
 The parabola opens down.
 The
vertex of the parabola is (25/3, 339/5).
 The
axis of symmetry is x= 25/3.

When
a=0.3, b=7.8, c=2
(a,b,c<0) 


When
a=0.3, b=7.8 and c=2,
 The parabola opens down.
 The
vertex of the parabola is (13, 48.7).
 The
axis of symmetry is x= 13.

Back to the Top 
2. Fix two of the values for a, b, and c. Make at least 5 graphs on the same axes
as you vary the third value.
How do you change the equations to explore other graph?


Graph of
when c varies:



As you see,
'c' is yintercept of the graph.
 As c increases, the graph moves up.
 As c decreases, the graph moves down.

Graph of
when b varies:


As you see,
'b' determines xcoordinate of the vertex of the graph with a.
 As b increases, the graph moves to the right.
 As b decreases, the graph moves to the left.

Graph of
when a varies:


As you see,
'a' determines whether the graph opens up or opens down and also determines the width of the graph.
 If a is positive, the graph opens up.
 If a is negative, the graph opens down.
 As a increases, the graph is getting thinner.
 As a decreases, the graph is getting wider.

Conclusion 
 'a' determines whether the graph opens up or opens down and also determines the width of the graph.
 If a is positive, the graph opens up, and if a is negative, the graph opens down.
 As a increases, the graph is getting thinner, and as a decreases, the graph is getting wider.
 'b' determines xcoordinate of the vertex of the graph with 'a'.
 As b increases, the graph moves to the right, and as b decreases, the graph moves to the left.
 'c' is yintercept of the graph.
 As c increases, the graph moves up, and as c decreases, the graph moves down.

Back to the Top 
4. Interpret your graphs. What happens to
(i.e. the case where b=1 and c=2) as a is varied?
Is there a common point to all graphs? What is it?
What is the significance of the graph where a=0?
Do the similar interpretations for other sets of graphs.
How does the shape change?
How does the position change?


Graphs of
when n is in [10, 10] 



The common point
to all graphs 

 These are the graphs of and when a=4.
 The vertex of the graph of is (0.125, 2.0625).
 Since a is 4 and is greater than 1, it is thinner than .
 Both graphs pass (0, 2).
 These are the graphs of and when a=0.05.
 The vertex of the graph of is (10, 7).
 Since a is 0.05 and is less than 1, it is wider than .
 Both graphs pass (0, 2).
 These are the graphs of and when a=0.
 There is no vertex because the graph of is a straight line.
 Both graphs pass (0, 2).
 These are the graphs of and when a=0.08.
 The vertex of the graph of is (6.25, 1.125).
 Since a is 0.08 and is less than 1, it is wider than .
 Both graphs pass (0, 2).
 These are the graphs of and when a=8.
 The vertex of the graph of is (0.0625, 1.96875).
 Since a is 8 and is greater than 1, it is thinner than .
 Both graphs pass (0, 2).

The siginificance
of the graph
where a=0? 

 These are the graphs of and when a=0.
 When a=0, the graph of is a straight line because the equation would be y=bx+c, which is a linear equation.
So, to be a parabola, 'a' should not be '0'; otherwise, it would be a straight line.
 Even though the graph is a straight line, the graph of when a=0 still passes (0, 2).

Conclusion 

 'a' determines whether the graph opens up or opens down and also determines the width of the graph.
 If 'a' is positive, the graph opens up.
 If 'a' is negative, the graph opens down.
 When a increases, the graph is getting thinner.
 When a decreases, the graph is getting wider.
 When a=0, the graph is a straight line because y=bx+c, which is a linear equation.
 Since c is yintercept, the graph always passes (0, 2). So, (0, 2) is the common point of the graphs.


Go Back to Top 
Go
Back to Michelle's Main page 
Go
Back to EMAT 6680 Homepage 
Copyright
@ Michelle E. Chung 