Signal flares were launched from two different boats. The red flare followed the function f(x)=-(x-15)^2+175 and the green flare followed the function f(x)=-x(

Question

. Signal flares were launched from two different boats. The red flare followed the function f(x)=-(x-15)^2+175 and the green flare followed the function f(x)=-x(x-25) , where x was the horizontal distance in meters and f(x) was the height in meters. Which flare went the highest in the air? Provide mathematical evidence to justify your answer.

Answer 1

Both of the equations describing the path of the flare are parabolas - because the coefficients of the x² terms are negative, each function looks like a little like an upside-down U.  Now, since the question asks us to find which flare went highest in the air, we need to find out which parabola is "taller" - or which upside-down U has the highest top.  We can do this by finding the vertex of the parabola - specifically, we're interested in the y-coordinate of each parabola's vertex.  Whichever parabola has the vertex with the highest y-coordinate value, that's the equation of the flare that went highest in the air.

It's easy to find the vertex of a parabola if it's in vertex form: y = a(x - h)² + k.  In this form, h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.  

Your first equation is already in vertex form, which is great!  That means that the y-coordinate of the vertex is 175 - so the red flare travels 175 meters in the air.

Your second equation isn't in vertex form yet, so we have to figure out how to do that.  We can rewrite f(x) = -x(x - 25) as f(x) = -(x² - 25x).  Hmm... we need this to look something like -(x - h)² - k.  So we can ask ourselves: if -x² - 25x were the first two terms of (x - h)², what would h be?  We know from long practice at factoring binomials that h would have to be 25 divided by 2, or 12.5; and (x - 12.5)² is x² - 25x + 156.25.  This allows us to rewrite our equation as:

 

f(x) = -(x² - 25x + 156.25) + 156.25 = -(x² - 12.5) + 156.25.

 

So the green flare traces out a parabola that hits its highest point at 156.25 meters.  Since the red flare went 175 meters in the air, it turns out that the red flare went higher in the air than the green flare!  And we have the mathematical evidence to justify the answer.  This kind of strategy can help you solve all kinds of problems having to do with parabolic trajectories in physics or parabolas in mathematics.