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Finding the Vertex of a Parabola

Parabolas are graphs with equations of the form , with .  Parabolas have a highest or a lowest point (depending on whether they open up or down), called the vertex.  Parabolas can model many real life situations, such as the height above ground of an object thrown upward, after some period of time, or perhaps the area of any rectangle with a width of, say, five inches less than its length.  We might wish to find, for example, the maximum height an object might reach, or the minimum area that such a rectangle described above can have.  The vertex of the parabola can provide us with that information, so we need to be able to find its coordinates.

Each parabola has a vertical line of symmetry that passes through its vertex.  Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts of a parabola.  We can find the x-coordinate of that midpoint, and consequently, the x-coordinate of the vertex, by taking the average of the two solutions to the quadratic formula since this formula gives precisely the two x-intercepts of a parabola.

Let’s consider f(x) = x2 – 4x – 5.  If we use the quadratic formula, recall that

x =  , where in this example, a = 1, b = -4, and c = -5.
Using the formula, we get x-intercepts of x = .
Note that this gives us:
=
=  , or 2  3.

This means that we start at 2 and add 3 to get one intercept, and subtract 3 from 2 to get the other intercept.   This means that 2 is the midpoint of the two intercepts (which happen to be 5 and  -1).

If we find the y-value that goes with the x-value of  2,  we’ll have the y-coordinate of the vertex.  , or -9.  Thus the vertex of our parabola is (2, -9).

Where did we get that number 2?
When we used the quadratic formula, it was obtained from .   This means that for any , then  will give us the x-coordinate of the vertex.  We need only to find that coordinate, and then find the y-coordinate that goes with it by using that value for x in our equation for f(x).

Example1

Find the vertex of the parabola given by the equation:

Solution 1

= , or 2

, or 3

The vertex of this parabola is (2, 3).

Example 2

Find the vertex of the parabola given by the equation:

Solution 2

= , or  -3.

, or -5

The vertex of this parabola is (-3, -5).

Example 3

Find the vertex of the parabola given by the equation:

Solution 3

= , or 4

, or 96

The vertex of this parabola is (4, 96).