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**Exploring Parabolas**

**By**

** Princess Browne **

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Try several graphs of y = ax^{2}

On the same axes (i.e. use
different values of a)

I will start by graphing the
equation of y = x^{2}

The graph of y = x^{2 }is
a parabola that is concave upward.

LetŐs look at few more graphs
where a is greater than 1.

Purple

Orange

Blue

Green

Teal

From
the graphs above, we can see that the graph is positive or concave upward when
a is positive. Each parabola share the same vertex point. The graph gets
skinnier or closer to the y-axis as a increases. The parabola does not have a
constant slope. If we look at the graph
of y= x we will notice that as x increases by 1, starting with x =
0, y increase by 1, 3, 5, 7, É As
x decreases by 1 starting with x = 0, y still increases by 1, 3, 5, 7,É
Look at the table below for better understanding.

X |
1 |
2 |
3 |
4 |

y |
1 |
4 |
9 |
16 |

Find slopes using (1,1),
(2,4), (3,9) and (4,16)

(y2-y1)/(x2-x1) = (4-1)/
(2-1) = 3

(y4-y3)/(x4-x3) =
(19-9)/(4-3) = 7

Using the models above we
will predict that when a is less than 1 the graph will get wider than the
original graph. LetŐs see our prediction of the graph when a is less than 1.

Purple

Red

Blue

Green

When a is less than 1 the
graph is larger than the graph of y=x^{2} and the points are moving
away from the vertex. The graph is closer to the x-axis and the graph is always
a parabola. Next, letŐs see what will happen to the graph when a is a negative
number.

Purple y
= -x^{2}

Orange y =
-2x^{2}

Blue y
= -3x^{2}

Green y
= -4x^{2}

Teal y
= -5x^{2}

The graph is still a
parabola. The parabola is negative or concave downward symmetric to the y-axis.
As a increases the graph gets skinnier. The graph is concave up (along the
y-axis) when a is positive and concave down (along the y-axis) when a is
negative. The positive graph has a minimum point; whereas the negative graph
has a maximum point. Now we will add a constant c to the equation and see what
will happen to the graph. The graph below is the graph of ax^{2 }+ c.

Purple

Orange

Blue

Green

Adding or subtracting the constant
c will shift the graph along the y-axis. When c is added to the equation the
graph is above the x-axis and moves up along the y-axis. When c is subtracted
from the equation the graph is below the x-axis and moves down along the
y-axis. For example: when c is 2, the graph will move up 2 units from the graph
of y = x^{2}. The vertex of y = x^{2} is (0,0) and the vertex
of y = x^{2}+ 2 is (0,2).

Finally, we will compare the
graph a(x-b)^{2} + c with the graph of ax^{2} + c.

Teal y
= x^{2} +1

Purple y
= (x-2)^{2} + 1

Red y
= (x-3)^{2} + 2

Blue y
= (x+2)^{2} + 1

Green y
= (x+3)^{2} + 2

The graph of y = (x-2)^{2}
+ 1 will shift the graph two units right from the graph of y = x^{2} +1
and the vertex is (2,1). The graph of y = (x+2)^{2} + 1 will shift the
graph two units left from the graph of y = x^{2} +1 and the vertex is
(-2,1). The equation of y = x^{2} +1 will shift the parabola up one
unit from the vertex or along the y-axis. Whereas, the equation of y = (x-2)^{2}
will shift the parabola two units right of the vertex or along the x-axis.