 In this video, we provide the solution to question number 14 from the practice final exam from math 1050 We're given a quadratic function f of x equals negative 2x squared minus 8x minus 2 when we have to find the vertex of This quadratic function. There's essentially two strategies that one could take here I'm going to implement both of them here. So one option is you can just complete the square So take out the negative 2 from the x's cell is behind x squared, and then you have a plus 4x Leave a spot for your guest of honor, then you have a minus 2 right there Then to identify the guest of honor, we're going to take this middle coefficient here 4 We take half of that which is 2 we then square that which is 4 So we add in a 4 then we have to subtract 4 times negative 2 where this negative 2 is the coefficient in play right there so then the The the the things inside of the parentheses the x square plus 4x plus 4 should be a perfect square trinomial It'll factor as x plus 2 quantity squared and then over here You have a negative 4 times negative 2 that's a positive 8 minus 2 is going to be a positive 6 So this is now in the vertex form from this form We can very quickly see that the Switching the sign here because we should have an x minus h inside of the square So that's going to give us a negative 2 that this is the y-coordinate of the vertex. So you get negative 2 comma 6 That would then be choice a as The correct answer. So that's what you would get if you solved it by completing the square But some of us like an alternative approach the idea that h equals negative to excuse me negative b over 2a Use this formula if you plug in The b and the a like so you would end up with a negative negative 8 over 2 times negative 2 So the top is positive 8 in the denominator You get negative 4 that simplifies to be negative 2 like we observed over here and over here So finding h is pretty easy this way, but then we have to find k to find k We're just going to evaluate the function at h. So we plug in Negative 2 into the formula right there For which then we can evaluate that as negative 2 times negative 2 Squared minus 8 times negative 2 minus 2 like so try to simplify this thing Negative 2 squared is positive 4 times the negative 2 gives you a negative 8 6 sorry 8 times 2 is 16. It's a double negative. So it's a positive 16 like so then you get a negative 2 8 take sorry 8 to take away from 16 gives you a positive 8 minus 2 gives you another 6 And so I gave us the value you're looking for there So that works and then one other strategy I'll mention here that if you don't like these tedious a function evaluations You can also use synthetic division which might be a little bit cleaner because it does the evaluation for you negative 2 negative 8 negative 2 here You plug in the negative 2 that we wanted to do run through the calculation bring down the negative 2 Negative 2 times negative 2 is positive 4 minus 8 is negative 4 times negative 2 It's gonna be a positive 8 minus 2 is 6. So again, you found the number 6 again on the synthetic division I think it's a little bit cleaner than just the function evaluation. You can do that with polynomials in general But however you want to do this calculation whether you want to complete the square Which is actually the strategy. I do recommend. I think it's gonna be the cleanest calculation and For this one you just complete the square, but you can also use formulas the h formula negative Over 2 a we like a lot. We can get h pretty quickly there But as there are multiple answers that involve the correct h we do have to figure out k For which then we have to evaluate the function at the h value I do recommend synthetic division the arithmetic will be easier there But you can just do the traditional plug-and-jug that'll work for you as well