Quadratic Word Problems from Standard Form

Question

“A football player kicked a 41-yard punt. The path of the ball was modeled by y = -0.035x2 + 1.4x +1, where x and y are measured in yards. What was the maximum height of the ball?

The player kicked the football toward midfield from the 18 yard line. Over which yard line was the ball when it was at its maximum height?”

Answer 1

For a quadratic formula it is the form ax^2 + bx + c. Quadratic formulas make graphs called parabolas. A parabola is shaped like a "U" and if a < 0 then it shaped like an upside down "U". The bottom of the "U" is called the vertex. This is where the min or max of a quadratic formula exists. The formula for calculating the vertex is x = -b/(2a)

For our equation -0.035x^2 + 1.4x + 1

a =-0.035

b = 1.4

c = 1

x = -(1.4)/(2*-0.035) = 20

So the vertex (or max of this function) happens when x = 20. After we find our x we plug it back into the original equation to find our y.

y = -0.035x^2 + 1.4x + 1

y = -0.035(20)^2 + 1.4(20) + 1

y = 15 yards

We know from earlier that our max height happens at x = 20. If we start at the 18 yard line and proceed towards midfield 20 yards we reach the 38 yard line.