The basic "parent" function for a parabola
is given as . The graph is shown below. Notice that the vertex of
the parabola (the lowest point) is at (0,0).
In this exploration, I want to see how the graphs
of compare when different values of d are used.
First, let's look at the graph of below. What do
you notice about this graph compared to the "parent" graph? Has
the position changed? Has the shape changed? Where is the vertex of the
Did you notice the position of the parabola changed
by moving down 2 units? The vertex is now at (0,2). However, the shape of
the graph compared to the parent has not changed.
Let's look now at the following graphs:
is the graph in purple,is
red, is blue
and is in green.
What is happening to the graphs?
As you can see, the position of the graph is moving,
however the shape is not changing. Can you make a conclusion about what
is happening as we subtract a number from ?
When we add a value to ?
As you should conclude, when we add to the graph shifts upward the number of units added and
subtracting from , causes the graph to
shift downward that many units.
Now let's explore what happens when we add or subtract
to the quantity of . As stated earlier,
let's look at different values of d, given the equation .
Do you have any thoughts at this time? Let's see
if you are correct!
The graphs of the following equations are given
below when d = 2, 4, -2, and -4.
Notice where the vertex of each parabola is located.
What conclusion can you make?
Through this exploration, you should have discovered
that when positive values of d are given, that the graph shifts to the right
the number of units d. When d takes on negative values, the graph shifts
left the number of units d.
When given an equation for a parabola, where 1
is the coefficient of x, you should be able to state the vertex of the parabola
prior to graphing.
Try stating the vertex of each parabola given below.
Look below to see if you were correct. Do the graphs
match up with what you thought would happen?
Good luck in your next adventures with parabolic