Question

The height of water shooting from a fountain is modeled by the function f(x) = −0.2x2 − 2.8x − 5.4 where x is the distance from the spout in feet. Complete the square to determine the maximum height of the path of the water.
A: −0.2(x + 7)2 + 5.4; The maximum height of the water is 7 feet.
B: −0.2(x + 7)2 + 5.4; The maximum height of the water is 5.4 feet.
C: −0.2(x + 7)2 + 4.4; The maximum height of the water is 4.4 feet.
D: −0.2(x + 7)2 + 4.4; The maximum height of the water is 7 feet.

Answer 1

Start by setting the function equal to 0 and then moving the -5.4 over to the other side of the equals sign.  $-.2 x^{2} -2.8x=5.4$.  The first rule for completing the square is that the leading coefficient be a +1.  Ours is a -.2.  So we need to factor it out. $-.2( x^{2} +14x)=5.4$.  Now we will 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.  So we add 49 in to the left side just fine, but we cannot forget about that -.2 hanging around out front as a multiplier.  What we have actually "added" in is -.2*49 which is -9.8.  Now here's what we have after all that: $-.2( x^{2} +14x+49)=5.4-9.8$.  In that process, we have created a perfect square binomial on the left.  Along with expressing that binomial we will do the math on the right: $-.2(x+7) ^{2} =-4.4$.  Now we will move the -4.4 back over by addition, and it will then be apparent as to what our vertex is.  The y coordinate of the vertex will give us the max height of the water.  $-.2(x+7) +4.4=y$.  As you can see, our work matches choice C from above.