Finding the vertex of a quadratic function

Every parabola has a vertex. Depending on the orientation of the parabola, it will always be the highest or lowest point on the parabola. Here is a picture of two parabolas graphed on the same coordinate plane with the vertices labeled.

The first thing you should remember is that the vertex is a point. And points look like this (3,5) Therefore you have to find two coordinates, the x-coordinate and the y-coordinate.

Step 1: Finding the x-coordinate of the vertex
If you are given a quadratic equation, say . All you have to do is note a,b, and c.
a = -5
b = 2
c = -5
And then use a little formula. to calculate the x-coordinate of the vertex.
In our example, is . Simplifying, which is 1/5. So we know this much about our vertex now, it is a point (1/5, ?)


Step 2: Finding the y-coordinate of the vertex
One tends to forget this, but whenever you have one coordinate of a point, and you know the equation you can always plug the coordinate you know into the equation and find the other. So in our case, we know the x-coordinate, we are going to plug it into our quadratic function, and find out our y-coordinate. In step 1 we figured out that the x-coordinate is 1/5 and we also have our function . So, watch the magic, we just plug in 1/5 for every "x" in our function: . Now we simplify the right side,

some more

and some more

and...

finally!

There's our y-coordinate!
So our vertex is (1/5, -24/5)

Here is a little demo I did from a question in the homework. (Watch out there is an embarrassing bit of subtraction in there. The y-coordinate should be -10, not -18.)


Also see:
-intro to 11.6
-the direction/orientation of the parabola
-the line of symmetry
-the y-intercept
-the x-intercepts