A quarterback tosses a football to a receiver 40 yards...

A quarterback tosses a football to a receiver 40 yards downfield. The height of the football, f(x), in feet, can be modeled by $$ f(x)=-0.025 x^{2}+x+6 $$ where x is the ball’s horizontal distance, in yards, from the quarterback. a. What is the ball’s maximum height and how far from the quarterback does this occur? b. From what height did the quarterback toss the football? c. If the football is not blocked by a defensive player nor caught by the receiver, how far down the field will it go before hitting the ground? Round to the nearest tenth of a yard. d. Graph the function that models the football’s parabolic path.

A quarterback tosses a football to a receiver 40 yards downfield. The height of the football, f(x), in feet, can be modeled by
$$
f(x)=-0.025 x^{2}+x+6
$$
where x is the ball’s horizontal distance, in yards, from the quarterback.
a. What is the ball’s maximum height and how far from the quarterback does this occur?
b. From what height did the quarterback toss the football?
c. If the football is not blocked by a defensive player nor caught by the receiver, how far down the field will it go before hitting the ground? Round to the nearest tenth of a yard.
d. Graph the function that models the football’s parabolic path.

Key Concepts

Graphing Parabolic Functions

Graphing the quadratic function involves plotting key features such as the vertex, y-intercept, and roots, and then sketching the parabolic curve. This visual representation aids in understanding the behavior of the projectile’s trajectory, including its maximum height and distance covered.

Roots of a Quadratic Equation

Finding the roots (or zeros) of a quadratic function involves solving the equation when the function's value is zero. In the case of projectile motion, the roots denote the points where the projectile reaches ground level, providing critical information about the range or distance traveled.

Y-Intercept

The y-intercept of a quadratic function is the point where the graph crosses the y-axis, which corresponds to the initial value of the function when x = 0. In projectile motion, this represents the height from which the object is launched.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on the graph, depending on whether the parabola opens downward or upward. In the context of projectile motion, the vertex gives the maximum height and the corresponding horizontal distance. It is calculated using the formula -b/(2a), where a and b are coefficients from the quadratic function.

Quadratic Functions

Quadratic functions describe parabolic relationships and are represented by a polynomial of degree two. They are used to model the trajectory of objects under the influence of gravity, where the graph of the function is a parabola. This type of function is key in analyzing the motion of projectiles such as a football in flight.

Transcript

00:01 So we're given a model for the height of the ball.

00:04 This is our height in feet.

00:11 And this is x is the distance away from the quarterback.

00:14 So negative 025x squared plus x plus 6.

00:21 And we want to find what the maximum height is of the ball.

00:25 And we also want to find how far that is away.

00:27 And i know this is a parabola that has a y intercept at 6.

00:35 And then the ball is going to be thrown and is eventually going to hit down.

00:40 I know it curves downward because i have a negative lead coefficient.

00:43 We want this point.

00:46 And i didn't draw this too symmetrically, but we know when he tosses that ball, it's going to form a parabola, and we want that vertex, basically.

00:55 And so the vertex is going to take place where this axis of symmetry is, which is at negative b over 2a.

01:04 So let's find what negative b over 2a is.

01:06 And that will be the x coordinate for what that peak is.

01:11 So we have negative b over 2a.

01:19 And let's multiply, divide that out.

01:22 So i have negative 1 divided by left parenthesis 2 times negative 0 .025.

01:31 And when i do that, i find 20 feet.

01:35 So 20 feet out will be what that maximum height is.

01:39 And then we need to find what f of 20 is.

01:42 So we need to plug it in.

01:43 So i'll do that.

01:45 I have negative 0 .025 times 20 squared plus 20 plus 6...