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Contributor: Michelle Haver. Lesson ID: 13638
Quadratic functions are useful in a variety of situations. Let's find out how to determine the maximum or minimum of these functions by completing the square.
You and your classmates are building model rockets to launch at school one day. On launch day, your teacher takes you to the roof of the school and has everyone set up their rockets.
After watching your rockets launch in the air, your teacher writes the following equation on the board:
h(x) = -16x^{2} + 96x - 44
Your teacher asks you to use the given equation, which gives the height of the rocket when x seconds have passed, to calculate the maximum height the rocket will reach.
To answer your teacher's question, let's consider a sketch of the situation:
*model rocket parabola
The graph of a quadratic function has the shape of a parabola. The maximum or minimum of a quadratic function is called the vertex. In a graph, this is seen as the highest point or the lowest point!
There's a special way we can write a quadratic expression, which will directly tell us the vertex. This is called the vertex form:
(x - h)^{2} + k
In this form, the vertex (maximum or minimum) is given by the ordered pair (h,k).
Now, let's figure out how to rewrite a quadratic expression in vertex form. Consider the quadratic expression x^{2} + 4x + 5.
We can first look at the vertex form and expand it:
(x - h)^{2} + k | = | (x - h)(x - h) + k | |
= | x^{2} - hx - hx + h^{2} + k | ||
= | x^{2} - 2hx + h^{2} = k |
One way to think about this is to match up the terms and their coefficients. This gives us a way to solve for the values of h and k:
x^{2} + 4x + 5 | = | x^{2} - 2hx + h^{2} + k | |
x^{2} | = | x^{2} | |
4x | = | -2hx | |
5 | = | h^{2} + k |
We can now solve for h and k:
4x | = | -2hx | |
4 | = | -2h | |
4 |
= | h | |
-2 | |||
5 | = | h^{2} + k | |
5 - h^{2} | = | k |
Using this information, we can rewrite the quadratic expression x^{2} + 4x + 5 in the following way:
(x - (-2))^{2} + (5 - (-2)^{2})
(x +2)^{2} + 1
The process of rewriting a quadratic expression into vertex form is called completing the square.
This process can be a little confusing, so let's recap the steps you take by working through an example together.
On the slides bleow, rewrite the quadratic expression x^{2} + 8x - 2 in vertex form:
The phrase completing the square actually has a nice visual representation geometrically.
To see this method in action, check out Completing the square using algebra tiles - Demo from missnorledge:
All the examples we have seen so far have a leading coefficient of 1.
The method is almost identical to what we have seen before, but with one added step at the beginning.
See an example of completing the square when the leading coefficient is not 1 in Complete the square with a number in front (leading coefficient) from Simple Math:
Now, let's remember why we even care about the vertex form of a quadratic expression in the first place!
This form is special in that it directly gives the maximum or minimum point on the graph of the resulting function. For a quadratic function in vertex form, the point (h,k) is the vertex:
y = a(x - h)^{2} + k
Your teacher has asked you to determine how high the rocket goes if its height is modeled by the following equation, where x is the number of seconds which have passed:
h(x) = -16x^{2} + 96x - 44
We can use completing the square to rewrite the equation in vertex form to find the maximum point!
Once you've found the equation yourself, check your answer below.
It's the ordered pair (3,100)!
Think about what an ordered pair represents.
We write ordered pairs as (x,y) or (input, output). Therefore, 3 is representing an input value, and 100 is representing an output value.
Since the output of the function h(x) is the height of the rocket, 100 means the rocket is at a height of 100 feet.
Since the vertex gives the maximum point in this case, we know that the maximum height of the rocket is 100 feet.
We could also say how long it will take the rocket to reach that height! Since the input at that point is 3, it will take the rocket 3 seconds to reach its maximum height of 100 feet!
Rewriting a quadratic expression in vertex form gives us nice information about the maximum or minimum of the function it defines, but completing the square can take some time to understand and feel comfortable with.
The following videos may be useful and will show several more examples of how this process works. For extra practice, check out these out and try to work the problems along the way!
Move on to the Got It? section to see!