Manipulating The Parabola

Damarrio C. Holloway

EMAT 6680 Summer 2006

Dr. Wilson

 

 

The victim of my problem is the equation:

 

 

GRAPH 1

 

 

Graph 1 is the simple parabolic graph of my initial equation.  Now we will begin the manipulation.

 

 

 

 

 

 

 

i) Overlay a new graph replacing each x by (x - 4).

 

                    

 

GRAPH 2

 

Graph 2 is the graph when each x is replaced by (x-4).  The new graph shifts to the right with the same vertex as the original graph and the roots on the graphs are equidistant. 

 

 

 

 

 

 

ii)  Change the equation to move the vertex of the graph into the second quadrant.

 

                                              

 

 

 

By changing the c coordinate to +2, raises the vertex of the graph to the 2nd Quadrant. 

 

 

iii)  Change the equation to produce a graph concave down that shares the same vertex.

 

 

 

Changing the a to -a does create a concave downward parabolic function, but it will not necessarily create a common vertex as seen below in graph 4.  The first two equations are the same but the new equation is .  The graph does concave downward with a tangent point of y=4.

 

 

 

 

GRAPH 4

 

 

 

 

 

 

 

 

 

 

                    

 

  New Equation

 

GRAPH 5

 

 

 

Changing the terms of x in the original equation produced the second function yielding the purple graph.  Going back to the instructions in part i), I began changing the x terms to give a phase shift to the left instead of a right phase shift.  Keeping the c coordinate negative leaves the vertex in the bottom two quadrants, but changing the a coordinate to (-), creates a downward parabolic function.  In graph 5, we see the perpendicular bisector is the line x=-0.75, and the common vertex that they share is y=-5.125. 

 

 

 

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