Football player

Question

A football player throws a ball as far as he can.the trajectory of the football is given by the function h(t)=-x^2+2x+3,where h(t) is the height of the ball, in meters, and t is the time in the air, in seconds.

a)How high is the ball being thrown from

b)find the zeros of the equation. What do the zeros mean in the context of this scenario

c) what is the vertex of the equation what does the vertex mean in the context of this scenario

Answer 1

So the trajectory of the thrown football is a parabola, an arch.

Part a). when x= 0, that's when the ball is first being tossed. Plug x into the equation to solve for y (the height, or h(t) )

y = -x²+2x+3

y = -0² + 2 (0) + 3

y = 0 + 0 + 3

y = 3

Therefore at the beginning, when x = 0, y = 3, the height, h(t), is 3 meters. The football is being thrown from a height of 3 meters.

Part b)

"zeros" is another word for "root", "solution", "x-intercept". It's where the parabola crosses the x-axis. In the context of the thrown football, it is the point where the football would be thrown from ground level, and where it lands on the ground.

To find the zeros of a quadratic equation, you need to factor it:

y = -x²+2x+3

y = - (x - 3) (x + 1)

Set = to 0

0 = - (x-3) (x + 1)

x = 3

x = -1

This means that at 3 meters away from the player, the ball lands. The other zero is at -1 meters. In this context, it's impossible to be -1 meters away, but we know that our player is throwing from a height of 3 meters, standing on a ladder or something. If the ladder was taken away, he would have to stand back another 1 meter to maintain the same trajectory (and he'd have to put in more strength in his throw, too!)

So now for part c). The vertex, or maximum, is the highest point on the arch. The formula needed for that is the line of symmetry, x = -b / 2a. If you think about it, a parabola is symmetrical. And if you were to cut it in half, that line of symmetry would be at the tip of the parabola. So that's the vertex, the maximum, the highest point on the parabola.

The equation of your parabola is h(t)=-x²+2x+3. a, b, and c are the coefficients (the numbers in front) of each term. So a is -1, b is 2, and c is 3. a is -1 because the first term is -x² and there is an understood 1 in front of the -x².

a = -1

b = 2

c = 3

Plug into the line of symmetry formula:

x = -2 / 2 (-1)

x = -2 /-2

x = 1

This means that the line of symmetry can be drawn at x = 1. Now use the fact that x= 1 to find the y-value of the vertex. (Remember that h(t) is the same thing as your y-value.)

y = -x²+2x+3

y = -(1)²+ 2 (1) + 3

y = -1 + 2 + 3

y = 4

Therefore, the coordinates of the vertex are x =1 and y = 4 --> (1, 4).