**PARABOLA**

In this presentation, I will construct graphs
for the parabola

for different values of a, b, and c. (a,
b, c can be any rational numbers).

Before plotting the graphs, I'd like to follow
a different approach: **COMPLETING TO SQUARE**. In this way we can see __how far a parabola is being translated from
the origin__. We can even rotate a parabola! But for
the moment, let's only translate it:

y = a x ^ 2 + b x + c = a [ x ^ 2 + ( b /
a ) x + (c / a) ] = a ( x - r ) ^ 2 + k ,
where , r = - b / 2a , and , k = ( 4 a c
- b ^ 2 ) / 4 a

With this notation, r = - b / 2a is the **symmetry axis** of the parabola , and , k represents its
**extremum value**. Now the parabola has the form **y = a ( x - r ) ^ 2 + k **which can also be written as ( y - k ) =
a ( x - r ) ^ 2 . Translation procedure is
complete! We translated the parabola to a
new coordinate system where the origin is
( r , k ). This new origin is nothing but
the **extremum point** (or **vertex**) of the parabola. The following plots summarize
this paragraph. __When a > 0 , the parabola looks up , and
when a < 0 , it looks down.__

**CASE ANALYSIS**

**(1) r = 0 => - b / 2a = 0 => b = 0**

In this case, the extremum point (vertex)
of the parabola is on the y - axis , **AND** , the curve is symmetric with respect to
the y - axis:

**(2) k = 0 => ( 4 a c - b ^ 2 ) / 4 a =
0 = > 4 a c = b ^ 2**

In this case, the discriminant of the 2nd
degree equation is zero. Equivalently, y = a ( x - r ) ^
2 . Therefore, y = 0 has __only one real solution__, and that solution is x = r . From this,
we can deduce that the point ( r , 0 ) is
the extremum point (vertex) of the parabola
**AND** it is on the x - axis : __THE PARABOLA IS TANGENT TO THE X - AXIS__.

**(3) r = 0 AND k = 0 => - b / 2a = 0 AND
b^2 - 4ac = 0 => b = 0 AND b^2 = 4ac**

In this case, the conditions in case 1 and
2 are both satisfied. This results in y =
a x^2. Therefore, the extremum point (vertex)
of the parabola is at the origin ( 0 , 0
). If a > 0, then the parabola looks up.
If a < 0 , it looks down.

**(4) If | a | increases, then the arms of
the parabola narrow. They get close to each
other.**

What are the possibilities when the parabola
has the form ? Then we have slightly different cases
now:

* The point ( 0 , c ) is the y - intercept
of the parabola

* If b = 0 => y - axis is the symmetry
axis of the parabola

* If c = 0 => One of the arms of the parabola
passes through the origin

* If b = 0 AND c = 0 => The extremum point
is at the origin

These slightly different cases can be seen
in the figure below:

In this way, I covered all the possible cases starting from a different approach. The following little tables summarize the cases I analyzed:

**(i) When the parabola has the form y = a
( x - r ) ^ 2 + k
**

r = 0 | The extremum point of the parabola is on the y - axis AND the parabola is symmetric with respect to y - axis |

k = 0 | The parabola is tangent to the x - axis |

r = 0 AND k = 0 | The extremum point of the parabola is at the origin |

| a | increasing | The arms of the parabola narrow. |

**(ii) When the parabola has the form **

What is c ? | The point ( 0 , c ) is the y - intercept of the parabola |

b = 0 | y - axis is the symmetry axis of the parabola |

c = 0 | One of the arms of the parabola passes through the origin |

b = 0 AND c = 0 | The extremum point is at the origin |

__APPLICATION__

Can we determine parabola's equation from
its graph?

__1st Solution:__

y = a ( x + 1 ) ( x - 4 ) and we can also
see from the figure that c = - 4 . Therefore
the point ( 0 , - 4 ) satisfies y = a ( x
+ 1 ) ( x - 4 ). After substitution, we get
- 4 = a ( 1 ) ( - 4 ) => a = 1 => y
= ( x + 1 ) ( x - 4 ) => y = x^2 - 3x
- 4.

__2nd solution__

y = a x^2 + b x + c AND c = - 4 => y =
a x^2 + b x - 4. From the graph, (-1,0) and
(4,0) satisfy the equation. Substitution
gives:

x = -1 => a - b - 4 = 0 => a - b =
4

x = 4 => 16 a + 4 b - 4 = 0 => 4 a
+ b = 1

Solving these two equations simultaneously,
we get a = 1 and b = -3 and hence y = x^2 - 3x - 4.

__3rd solution__

c = - 4

Product of the roots = c / a => (-1) (4)
= c / a => a = 1

Sum of the roots = - b / a => -1 + 4=
- b / 1 => b = -3

Hence y = x^2 - 3x - 4.