You are attending a professional baseball game. The batter hits the ball into right field. You notice the ball, as it is hit, forms a parabolic path from the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation y=−142(x−50)2+60

Question

𝑦
=
−
1
42
(
𝑥
−
50
)
2
+
60
, where x is the distance from home plate, and y is the height in the air as the ball travels in the air. Interpret the following key feature of the parabola in terms of the context.

Answer 1

The quadratic equation you provided represents a parabolic path of the baseball. The equation is given in the vertex form, which is:

\[ y = -\frac{1}{42}(x - 50)^2 + 60 \]

In this form, \( (x - 50)^2 \) indicates a horizontal shift of the parabola, and the vertex, which represents the highest point of the parabola, is located at \( (50, 60) \).

Key Feature: The Vertex

Interpretation:

1. Vertex Coordinates (50, 60):

   • x-coordinate (50): This indicates the horizontal distance from home plate to the point where the baseball reaches its maximum height. In this context, the batter hits the ball and it reaches its highest point when it is 50 units away from home plate.

   • y-coordinate (60): This represents the maximum height of the baseball. It tells us that the highest point the ball reaches in the air is 60 units (feet) above the ground.

Contextual Summary:

In simple terms, this means that as the batter hits the ball and it travels through the air, it reaches its peak height of 60 feet at a distance of 50 feet from home plate. After this point, the height of the ball will start to decrease as it continues to move away from home plate until it eventually lands on the field. This information can help assess the ball's trajectory and potential landing area in the outfield.