The Vertex

The shape a quadratic equation makes when graphed is called a parabola.  For a quadratic equation, parabolas will always “open up,” or “open down,” as seen below.  The sign of the leading coefficient, which we call a, can tell you the direction. 

Quadratic equations are generally written in one of two fashions. There is standard form, which is ax² + bx + c, and there is vertex form, which is a(x – h)² + k.  (We will get discuss vertex form in greater detail in a future section.)  How you find the vertex’s coordinate depends on the form of the equation.  From vertex form, it is as easy as can be, it can be by visual inspection, like finding the y – intercept from standard form.

Vertex form again is y = a(x – h)² + k.  The vertex will be (h, k).  The only thing to be careful of is the sign of h.  Let’s see a few examples.

Finding the vertex from standard form is a little trickier.  Here we have to use a formula, which looks intimidating. We discussed in the previous section, but it deserves its own focus.  That way, it will not prove too difficult for you.

An example of standard form is y = 3x² – 2x + 4.  Here the coefficient a = 3, b = -2, and c = 4.  Here’s how you find the coordinate of the vertex.

Step 1:  The x – coordinate will be “the opposite of b divided by twice of a.”  In math, that would be written as follows.

Since we know the values of a and b, from just looking, we plug those values in.  Remember, these are coefficients, not the entire term.  In other words, the variable x is NOT part of a or b. It is a good idea to just write in the values, without simplifying the negative or reducing initially.  After you write it, then reduce and simplify the signs.

Here the signs simplify to positive, and the twos reduce.  The x – coordinate is 1/3. 

Step 2:  Since x is the input, to find the y – coordinate, all you have to do is plug in the x value into the original equation and do the math.

Just like before, plug it in, carefully.  Be neat, use lots of space.  A sure way to increase your mistakes is to be sloppy, write with a small size, and cram your work into a small space. 

Once you have plugged everything in, just follow the order of operations to do the work.  Square first, then reduce.  Then, combine like terms.  There’s a pattern that always happens here that makes life easy though.  Let’s do the first part of this simplification and then look at the pattern.

Square first, then reduce.

Let’s stop here.  Look at the first two terms, 1/3 and -2/3.  What do you notice about them?  Do you see that the second term has twice the “size” of the first term?  Sure, one is negative, which is smaller than the positive, but the size is twice as big.  That always turns out to be the case.  This can save you a ton of time.  When you subtract a number two numbers, one twice as big as the other, something easy happens.  Let’s look at a few example.  Notice the first number will always be half of the second, or, another way to say it, the second number is twice of the first.

Sometimes, the denominators are not the same, so it is hard to see that the numbers have that relationship.  Here are a few examples.

To see how this works, let’s get a common denominator and see the pattern.

When used carefully, that pattern can save you a lot of time, headache, and mistakes!

Looking at our initial example, we found that x was 1/3.  When we plugged that into the original equation, we ended up with this below.  Let’s finish it.

There are two ways to approach this.  You can either deal with these as improper fractions or as mixed numbers.  The choice is yours, do whatever is easier for that particular set of numbers.  But, be proficient with both cases!

In this example, the mixed number was far easier.  If you take 1/3 away from 4, you still have a full 3, plus the 2/3s.  But sometimes, the mixed number is trickier.

In the end, our vertex is

Let’s see that same exact thing, but in a condensed format where we don’t explain all of the little stuff.

So the vertex is in the first quadrant!  Since a is positive, the parabola opens up.  That means this parabola will not cross the x – axis.  It will not have any x – intercepts.

The Formula

The formula for the vertex looks really ugly.  But, we’ve seen it in action.  The formula is a coordinate.  You find the x – coordinate first, then plug it into the original equation to find the y – coordinate.  The formula uses function notation to show how to find the y – coordinate.  Here is the formula.

Here’s a diagram to help you understand how to use the formula.  This is helpful, especially if you remember what we’ve learned about the fractions.  Keep in mind that just reading and seeing won’t make it a solid understanding.  You’ll have to try quite a few on your own before you really “get it.”

Practice the problems, be patient with yourself.  You’ll get this and likely find that it looks far more complicated than it actually turns out to be!