Parabolas

By Courtney Cody

In this exploration, we will look at various graphs that have a different value of d for the following function:

Let us start by looking at the simplest case of the graph of  when d=0.  It is important to realize that although d can take on any number, it is always some numeric value and not a variable.  Therefore, d is a constant in the equation  and we want to see how varying d changes the graph of this equation.

Graph A

Now, when d=0 the equation yields a parabola centered about the y-axis and the vertex is at (0, 2), meaning the graph is shifted upward by 2 units.  Also, the parabola opens up.  We should expect all equations of the form  to result in this same parabola, and the only change will be determined by our value for d.

To see the affect that varying d has on the graph, let us observe the following set of graphs, which contain several equations of the form  with positive values for d.

Graph B

By only changing the value of d in the equation, the graph appears to have moved to the right.  The shape of the curve in Graph A has the exact same shape as the curves in Graph B.  It appears as though only the position of the curves has changed.  LetŐs examine the location of the vertices for these different curves.  Looking at the purple curve, which has an equation of , the vertex is (1, 2).  If we compare this to the vertex of the curve in Graph A, which has an equation of  and a vertex of (0, 2), we notice that the x-value of the vertex has increased from 0 to 1.  Now, if we look at the royal blue curve, which has an equation of , the vertex appears to be at (5/2, 2). Hence, as the value of d gets larger, the parabola simply shifts further to the right.  These values are precisely the values of d in the corresponding equations.  Also, having fractional values of d seems to have the same affect as the positive integer values of d, which is the x-value of the vertex is d.

LetŐs see if this is true for negative values of d.

Graph C

Now, by changing the value of d in the equation to be negative values, the graph appears to have moved to the left.  The shapes of the curves in Graph C have the exact same shape as the curves in Graph A.  Only the position of the parabolas has changed and they have shifted to the left of the y-axis.  If we look at the purple curve, which has an equation of, the vertex is (-1,2).  Also, if we look at the royal blue curve, which has an equation of, the vertex is (-5/2, 2).  Hence, as the value of d gets smaller, the parabola simply shifts further to the left.

Thus, no matter what the value of d is, the vertex of the parabola with equation  is always (d, 2).  Furthermore, the shape of the parabola is not affected by the value of d, only the location of the vertex is affected.