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 parabola now?

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:

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 functions!