Question
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 , 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.(1 point) Range: 0≤y≤
Answers
GPT-4o mini
In the context of the given quadratic equation \( y = -142(x - 50)^2 + 60 \), we can analyze the range of the function, which describes the height of the baseball as it travels in the air.
The vertex of the parabola given by the vertex form \( y = a(x - h)^2 + k \) represents the maximum height of the baseball during its flight. In this case, the vertex is at \( (50, 60) \), meaning that the maximum height (or apex) the ball reaches is 60 feet.
Since the equation is a downward-opening parabola (indicated by the negative coefficient of the squared term), the baseball will eventually land on the ground. Thus, the lowest height \( y \) can reach, which is when the ball hits the ground, is \( y = 0 \).
Therefore, the range of this quadratic function in the context of the baseball's flight is:
**0 ≤ y ≤ 60**
This implies that the ball will be at a height of 0 feet when it lands (in the outfield, on the ground) and will reach a maximum height of 60 feet during its flight.
The vertex of the parabola given by the vertex form \( y = a(x - h)^2 + k \) represents the maximum height of the baseball during its flight. In this case, the vertex is at \( (50, 60) \), meaning that the maximum height (or apex) the ball reaches is 60 feet.
Since the equation is a downward-opening parabola (indicated by the negative coefficient of the squared term), the baseball will eventually land on the ground. Thus, the lowest height \( y \) can reach, which is when the ball hits the ground, is \( y = 0 \).
Therefore, the range of this quadratic function in the context of the baseball's flight is:
**0 ≤ y ≤ 60**
This implies that the ball will be at a height of 0 feet when it lands (in the outfield, on the ground) and will reach a maximum height of 60 feet during its flight.