answer:
the answer is
all of these have max values because the parabolas have leading coefficients of -1. that means that they are upside down like a mountain, and mountains have high points, or max values. in order to find the max value we are asked to find we will have to complete the square on all of these, because completing the square puts a parabolic equation (quadratic) into vertex form. the vertex dictates the highest point or the lowest point on the graph. i'll do the first one (which isn't the answer) so you can see how to do it, then i'll do the correct one after that. the first thing to do is to set the quadratic equal to 0 and then move the constant over to the other side. that looks like this: . now, since the one very important rule for completing the square is that the leading coefficient has to be a 1 (ours is a -1), we have to factor out the negative like this: . the process is to take half the linear term, square it, and add it to both sides. our linear term is 14. half of 14 is 7, and 7 squared is 49. we add 49 into the parenthesis, but we cannot disregard that -1 sitting our front there, refusing to be ignored. it is a multiplier. what we have really added in is -1*49 which is -49. that is what we add to the right side: . we can easily do the math on the right to get -9, but on the left, what we have done all of this for is for the purpose of creating a perfect square binomial which is the h coordinate of the vertex. expressing the left side as this perfect square binomial gives us this: . now we can move the 9 back over by addition to get . the vertex, or max value here, is (7, 9). not what we are looking for. now let's move to the last one. following the same set of rules, . we factor out the negative to get . our linear term is 18. half of 18 is 9, and 9 squared is 81: . simplifying the right side and expressing the left as the perfect square binomial, we have . moving the 7 back over, gives us a vertex, or max value, of (9, 7). the last choice is the one you're looking for.
step-by-step explanation:
