Path of a baseball Assume a baseball hit at home plate...

Path of a baseball Assume a baseball hit at home plate follows a parabolic path having equation $$ y=-\frac{3}{4000} x^{2}+\frac{3}{10} x+3 $$ where $x$ and $y$ are both measured in feet. (a) Find the maximum height of the baseball. (b) Does the baseball clear an 8 -foot fence that is 385 feet from home plate?

Path of a baseball Assume a baseball hit at home plate follows a parabolic path having equation
$$
y=-\frac{3}{4000} x^{2}+\frac{3}{10} x+3
$$
where $x$ and $y$ are both measured in feet.
(a) Find the maximum height of the baseball.
(b) Does the baseball clear an 8 -foot fence that is 385 feet from home plate?

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two and are typically expressed in the form ax² + bx + c. They graph as parabolas, which can open upward or downward depending on the sign of the coefficient a. These functions are widely used to model situations involving constant acceleration, such as the motion of projectiles.

Vertex of a Parabola

The vertex of a parabola represents either its highest or lowest point, depending on whether the parabola opens downward or upward. For a downward-opening parabola, the vertex is the maximum point, making it essential for determining the peak of a projectile's path. The vertex’s x-coordinate can be found using the formula -b/(2a), after which substituting back into the equation provides its y-coordinate.

Function Evaluation and Substitution

Evaluating a quadratic function by substituting specific x-values into the equation is a key technique used to determine the value of y at various points along the curve. This process is particularly useful for verifying if a projectile reaches a certain height or clears an obstacle at a given horizontal distance.

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