Let's examine the quadratic function
and vary the values of d. In particular, we want to decide whether or not varying the values of d will change the general shape and/or position of the graph. We will begin by examining the function
which we will call the "parent" graph for our quadratic functions.
Notice that if d = 0, the rule would be
and the graph moves down the y-axis two units.
Again, our "parent" graph's vertex remains at the point (0,-2), the green parabola represents the function
and the parabola has its vertex at the point (1,-2). The blue parabola is a graph of the function
with a vertex of (2,2). It would appear as though the y-coordinate of
the parabola has remain unchanged, whereas the x-coordinate has moved one
unit to the right each time we increase d by 1. It follows that as d increases,
the graph moves d units to the right when d is positive.
Let's compare the "parent" with our new rule:
As before, our red parabola represents our "parent" and the green parabola represents our rule
Again, as we allowed d to be negative, the vertex of the parabola moved
d units to the left.
One may easily see from the graphs above that the general shape and direction
of the parabola remained unchanged, whereas the position of the parabola's
vertex along the x-axis changed depending on the chosen value of d. Indeed,
the parabola moves d units to the right if d is some positive real number
and the parabola moves d units to the left if d is some negative real number.