Jack throws a baseball. Its height above the ground (in...

Jack throws a baseball. Its height above the ground (in feet) is given by $$h(x)=-.0013 x^{2}+.26 x+5.5$$ where $x$ is the distance (in feet) from Jack to a point on the ground directly below the ball. (a) How far from Jack is the ball when it reaches the highest point on its flight? How high is the ball at that point? (b) How far from Jack does the ball hit the ground?

Jack throws a baseball. Its height above the ground (in feet) is given by
$$h(x)=-.0013 x^{2}+.26 x+5.5$$
where $x$ is the distance (in feet) from Jack to a point on the ground directly below the ball.
(a) How far from Jack is the ball when it reaches the highest point on its flight? How high is the ball at that point?
(b) How far from Jack does the ball hit the ground?

Key Concepts

Quadratic Functions

These are polynomial functions of the form f(x) = ax² + bx + c. Their graphs are parabolas, which open upward if a is positive and downward if a is negative. Quadratic functions are fundamental in algebra and model various physical phenomena, including the trajectory of projectiles.

Vertex of a Parabola

The vertex is the highest or lowest point on the graph of a parabola, depending on whether the parabola opens downward or upward. It can be found using the formula x = -b/(2a) for the x-coordinate, and substituting this back into the function gives the y-coordinate. This concept is crucial for finding the maximum height in projectile motion problems modeled by quadratic functions.

Solving Quadratic Equations

This involves finding the values of x that satisfy the equation ax² + bx + c = 0. Techniques include factoring, completing the square, and the quadratic formula. In the context of projectile motion, solving the quadratic equation helps determine when the projectile returns to ground level (i.e., when the height is zero).

Projectile Motion Modeling

Projectile motion is often modeled using quadratic functions when air resistance is negligible, describing how an object moves under the influence of gravity. The function represents the height of the projectile as a function of horizontal distance, where the vertex corresponds to the maximum height, and the zeros indicate the points where the projectile touches the ground.

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