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.