**Assignment #2**

**Explorations
in the Shifts**

**of Parabolas**

**by**

**Kimberly
Burrell**

**The following
will be an exploration involving both horizontal and vertical
shifts of a parabola given by the equation **

**.**

**The following
explorations were prepared on the Graphing Calculator software by
Pacifictech. **

**HORIZONTAL
SHIFTS**

**In order for the parabola to
shift directions horizontally, one must change the values of d in
the equation **

**.**

**In this equation the vertex is
denoted by (d, -2).**

**The graph below illustrates the
type of shift that occurs when d = 1, d = 2, d = 4, and d = 6.**

**From this graph, one can
observe that a positive value of d will result in a horizontal
shift to the right by the number of units, d.**

**The next graph demonstrates
the type of shift that occurs when d = -1, d = -2, d = -4, and d=
-6.**

**One can observe from the
above investigation that a negative value of d will result in the
graph shifting horizontally to the left, d units.**

**VERTICAL
SHIFTS**

**In order for the parabola
to be shifted vertically, one must change the values of k in the
equation**

**.**

**In this form the vertex is
(d, k). While performing this exploration d will remain zero.**

**The graph below
illustrates the type of shift that occurs when d = 2, d = 3, d
=5, and d = 7.**

**From this graph, one can
observe that a positive value of k will result in a vertical
shift upward by k units.**

**The next graph
demonstrates the type of shift that occurs when d = -2, d = -3, d
= -5, and d=-7.**

**One can observe from this
investigation that a negative value of k will result in a
vertical shift downward by k units.**

**CONCLUSION**

**By
changing the values of d, the parabola will be shifted
horizontally by d units. This shift will be to the right if d is
a positive number or to the left if d is a neagtive number.**

**By
changing the values of k, the parabola will be shifted vertically
by k units. This shift will be in the upward direction if k is a
positive number and in the downward direction if k is a negative
number.**

**These
changes do not effect the shape of the parabola.**

**EXTENSION**

**In order
to change the shape of the parabola one must change the
coefficient of x.**

**An integer
coefficient of x will make the parabola to become narrow or
stretch. A rational coefficient, which is not an integer, of x
will make the parabola become wider or shrink.**