5.5CompletingtheSquarefortheVertex
Having the zeros is great, but the other key piece of a quadratic function is the vertex. We can find the
vertex in a couple of ways, but one method we’ll explore here will be useful later when we solve quadratic functions
as well. Let’s start with the intuitive method for finding the vertex.
If we know that the zeroes of the function 2 15
are at 3 and 5, then we can find the vertex by going half-way
between the zeros. Half-way between 3 and 5 is 1, so
our vertex must be there. Next find 1 for the -value of the vertex.
Using the Zeros to Find the Vertex
1 1 21 15 1 2 15 16
So we see that the vertex is at the point 1, −16) which we got
from our zeros at = 3 and = −5. Let’s take a look at the graph to
confirm that this is true.
Zeros
Vertex
Vertex Form
While the previous method of finding the vertex is simple, completing the square has more applications and
will allow us to solve quadratics later on. Let’s start by exploring some quadratics written in a specific format.
Consider the following graphs and equations of functions. What do you notice about the equation and the vertex
of each function? Could you find the vertex just from the equation?
2 3 7
F 4 5
5 9
Vertex: 3, −7
Vertex: 4, −5
Vertex: 5, 9)
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What did you notice? First, you may have noticed that these quadratics are not written in standard form.
Take F() for example. In standard form it would be:
F() = ( + 4) − 5
F 4 4 5
F 8 + 16 − 5
F 8 + 11
Taking a quick moment should convince you that F will not factor with integers to get the zeros, so we
have to find the vertex somehow without using the zeros. To do so, F was written in a different way, factored
in a different way, so that we could see the vertex easier.
Secondly, you may have noticed that the vertex is directly in the equation written in this new form.
F = + 4 5
-coordinate of vertex will make these
parentheses zero. This is similar to
translation left or right in previous functions.
Vertex: 4, −5
-coordinate of vertex is listed
directly. This is similar to translation
up or down in previous functions.
In fact, this form of writing quadratics is called vertex form. Formally, vertex form looks like this:
= ? − ℎ + h
where the vertex is at , h
Now we can see why the vertex for the function 2 3 7 is at 3, −7 and the vertex for the
function 5 9 is at 5, 9). So how do we get a function in vertex form?
Completing the Square
To get a quadratic function in vertex form, we use the process known as completing the square. It’s called
completing the square because we take a quadratic function written in standard form and write it with a single
instance of the in the parentheses being squared. We complete it by making a perfect square trinomial. Let’s
walk through an example of the process.
Let’s start with our function F 8 + 11 and write it in vertex form. We already know it should
end up as F = + 4 − 5, so it’s just a matter of how to get it there. Start by looking at the and terms
in the quadratic. If the function were just F = + 8, how could we change the function so that it would
factor into two binomials that are the same?
Add and subtract the same amount so that the function
doesn’t change in value. We have only added zero.
F = + 8 = + 8 + 16 − 16
Use 16 because we can then factor into + 4 + 4
which gives me the middle term of 8.
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Now we can factor just the first three terms in our modified function as follows:
F = + 8 + 16 − 16 = + 4 + 4 − 16 = + 4 − 16
Factor just this part.
This tells us that F = + 8 is exactly equal to + 4 − 16. However, our original function was
not F = + 8. It was F = + 8 + 11. So we’ll just have to add eleven onto our F function to
make it equal to our original function.
F = + 4 − 16 + 11
F = + 4 − 5
Now we have our function written in vertex form. Let’s work through a few more examples.
= + 4
F = + 6 + 11
ℎ = − 10 + 2
= + 2 + 2 − 4
F = + 3 + 3 + 2
ℎ = − 5 − 5 − 23
= + 4 + 4 − 4
= + 2 − 4
Vertex: −2, −4
F = + 6 + 9 − 9 + 11
F = + 3 + 2
Vertex: −3, 2
ℎ = − 10 + 25 − 25 + 2
ℎ = − 5 − 23
Vertex: 5, −23
Did you notice that the number we add and subtract is always half the -term’s coefficient squared? In
other words, if a function is written in standard form as = + @ + +, to complete the square we add and
ž subtract AB . Notice that’s without an ? value.
What would we do if there is an ? value? First, factor out ? from the original function and then complete
the square with the remaining quadratic.
Why is , · the Vertex?
I’m so glad you asked! Very inquisitive. You may have noticed that the vertex gives us the maximum or
minimum of the quadratic. So why would the vertex be , h in the form h?
If we want the minimum of that function, what is the smallest value we could have? If we input then
we will make the parentheses zero giving the function an overall value of h. This is the smallest value that
we can get since anything besides zero squared would be a positive value. Therefore, h must be the minimum
value making the point , h the vertex.
A similar argument can be made for a maximum if the parabola is upside down for the function h. The largest value is when since anything else will be negative.
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Lesson 5.5
Find the vertex using any method.
1. 2 − 8
2. () + 4 − 4
3. () − 5 + 4
4. () + 4 + 10
5. = 2 − 12 + 6
6. = + 3 − 18
7. = − 4 + 8
8. = 2 − 12 + 16
9. = 3 + 9 + 3
10. = − + 5
11. = 3 + 9 + 6
12. = − 4 + 4
13. = + 8 − 4
14. = + 2 − 1
15. = + 6 + 3
16. = −2 + 8 − 4
17. = − 6 + 7
18. = + 2 + 3
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Do a quick sketch of the graph of each function by finding first finding the vertex and then filling out the /
chart or finding the zeros.
19. 6 1
20. 2 6
22. 8 12
23. 2 8
21. 6 8
24. 4 2
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