Question

The height of a hockey puck that is hit toward a goal is modeled by the function f(x) = −x2 + 8x − 10, where x is the distance from the point of impact. Complete the square to determine the maximum height of the path of the puck.

Answer 1

I'm going to try using the instructions from the following resource to "complete the square" and answer this question:
https://www.freecodecamp.org/news/how-to-complete-the-square-a-method-for-completing-the-square/#:~:text=EXAMPLE%201%3A%20Completing%20the%20square%201%20Separate%20The,9%2F4.%20...%205%20Take%20The%20Square%20Root.%20

Answer 2

First, let's separate the terms of the equation into two sides, by whether or not they contain the variable x. We can also multiply both sides by -1 to ensure that x^2 has a coefficient of 1:
x^2 - 8x + __ = -10

To find the missing constant, we can take the coefficient of x (8) and divide that by 2, then square the result. 8/2 is 4, 4 squared is 16. We'll add this to both sides of the equation:
x^2 -8x + 16 = 6
We can factor the left side into (x-4)^2, so our equation becomes:
(x-4)^2 = 6.
Then, take the square root of both sides:
x-4 = squareroot(6)
x = 4 + squareroot(6)

Answer 3

Now, I'm a bit rusty on this method, to the point that I had to follow the an online tutorial above, but I think this means that when x = 4+squareroot(6), the height equation is maximized. We can check this with some calculations or a graphing calculator. One moment please...

Answer 4

Ok, I was partly right and partly wrong. Turns out I made the question a little more complicated than it actually was.
In reality, the solution is as simple as completing the square of the left side of the equation. Once we get (x-4), we're done: that tells us that the height is maximized when x ==4.
So, to get the maximum height from the equation, we can plug in 4 for x

Answer 5

For reference, here's what the graph looks like when it's plotted out

Answer 6

That was plotted with desmos.com btw, very useful online graphing calculator.
Good for checking our answer, but it's also important to understand how we got there by completing the square of the equation