Question

A) given the function f(x) = −x 2 + 8x + 9, state whether the vertex represents a maximum or minimum point for the function. explain your answer.
b.rewrite f(x) in vertex form by completing the square.

Answer 1

Since this is a negative parabola, the curve will open downwards like a mountain.  A mountain has a peak or a top, aka a maximum.  Now let's complete the square to see what that vertex is.  Rule is that the leading coefficient has to be a +1.  Ours is a -1.  So we need to factor it out.  First though, set the equation equal to 0 then move the constant away from the x terms to the other side of the equals sign.  $-x^2+8x=-9$.  Factoring out the -1, $-1(x^2-8x)=-9$.  Now we are ready to complete the square.  Take half the linear term, square it, and add it in to both sides.  Our linear term is 8.  Half of 8 is 4, and 4 squared is 16.  So we will add 16 into the set of parenthesis, but don't forget that -1 hanging out front there, refusing to be ignored.  It is a multiplier.  Therefore, what we have REALLY added in was (-1)(16) = -16.  That's what gets added onto the right side to keep things in balance.  $-1(x^2-8x+16)=-9-16$.  We simplify on the right to get -25.  Now on the left we will write the perfect square binomial we created while doing this process:  $-1(x-4)^2=-25$.  Move the -25 over to the other side by addition and set it back equal to y: $-1(x-4)^2+25=y$.  We see now that our vertex is (4, 25) and that it is a maximum