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 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.

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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 y-intercept 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 x-coordinate 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 x-coordinate 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 y-intercept of the graph.

    - As c increases, the graph moves up, and as c decreases, the graph moves down.

 

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4. Interpret your graphs. What happens to

           Para_euq12

     (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

Para_equ13

when n is in [-10, 10]

    

 

The common point

to all graphs

 

  

Para_graph11

  • 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).

 

 

Para_graph12

  • These are the graphs of Para_equ15 and Para_equ12 when a=-0.05.
  • The vertex of the graph of Para_equ12 is (10, 7).
  • Since |a| is 0.05 and is less than 1, it is wider than Para_equ15.
  • Both graphs pass (0, 2).

 

 

Para_graph15

  • These are the graphs of Para_equ15 and Para_equ12 when a=0.
  • There is no vertex because the graph of Para_equ12 is a straight line.
  • Both graphs pass (0, 2).

 

 

Para_graph13

  • These are the graphs of Para_equ15 and Para_equ12 when a=0.08.
  • The vertex of the graph of Para_equ12 is (-6.25, -1.125).
  • Since |a| is 0.08 and is less than 1, it is wider than Para_equ15.
  • Both graphs pass (0, 2).

 

 

Para_graph14

  • These are the graphs of Para_equ15 and Para_equ12 when a=8.
  • The vertex of the graph of Para_equ12 is (-0.0625, 1.96875).
  • Since |a| is 8 and is greater than 1, it is thinner than Para_equ15.
  • Both graphs pass (0, 2).

 

The siginificance

of the graph

where a=0?

 

  

Para_graph15

  • These are the graphs of Para_equ15 and Para_equ12 when a=0.
  • When a=0, the graph of Para_equ12 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 Para_equ12 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 y-intercept, the graph always passes (0, 2). So, (0, 2) is the common point of the graphs.

 

 
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