Vertex Form and Absolute Value

In this discussion of Vertex Form, we are still talking about absolutes.  This information really starts in Algebra 1, the page entitled Absolute Value.  We are going to assume you understand that information and are ready to move onward to Part 2, otherwise click on the bold letters to end up at part 1.  An interesting game can be found at Absolute Value Game.

Why do we need Absolute Values?  

There are a lot of things that you don't know enough about to understand why they are important.  So, the real reason is that absolute values are used for a lot of different reasons in math and the sciences.  i can give a few examples, but there are a lot of other reasons.  One reason is that computers are kind of picky, you give them a negative number when they are expecting a positive - and your doomed.  There are a lot of situations where a negative number is unacceptable.  How far is it to New York?  How tall is uncle Bob?  How old are you?  What score did you get on that math test? - None of these should have answers in the negative.  

Vertex Form

The graph consists of two rays that form a V.  The corner point of that V is referred to as a Vertex.  Which brings us to a special form called the Vertex Form (we’ll talk about this more when we get into higher exponent functions. 

The absolute value vertex form is   f(x) = |x – h| + k

When written in this form, we know several things about the graph –

        a)      The graph has a vertex at (h,k)

        b)      The graph is V-shaped and if a > 0 the V opens up.  If                    a < 0 it opens down.

         c)       The graph is symmetric in the line x = k

The Vertex form is really handy if you want to easily identify the vertex (h,k) and the axis of symmetry k.

An Example, or two

We’ll start with an easy one f(x) = 5|x + 4| + 6. 

This graphs as a V with it’s vertex at (4,6)  and the axis of symmetry at x = 6 (the value of k).  In this case a is 5, h is 4, and k is 6. 

And now a little harder --

f(x) = a|x - h| + k

So, a sample problem might be f(x) = -2|3x + 7| -30.  We are unhappy!  This is not quite in the f(x) = a|x + 7| + k form, so we have to fix it.  We are multiplying x by 3, and that’s not good.  So, take a 3 out of the |3x + 7| part.  Would you agree that  -2(3)|x + 7/3| is the same as |3x + 7|? 

So our equation now becomes f(x) = -6|x + 7/3| - 30.  Not really pretty, but now our a is -6, our h is 7/3 and our k is -30.

We have an upside down V (because a is negative).  With a vertex at (7/3, -30), and a point of symmetry at x = -30.