Question

You are attending a baseball game. The batter hits the ball high into the air of the infield. You notice the ball, as it is hit, forms a parabolic path from contact with the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation y=−124x2+2x
, where x represents the horizontal distance and y represents the height of the ball. Interpret the following key feature of the parabola in terms of the context.(1 point)
Range: 0≤y≤

Answer 1

To interpret the range of the quadratic equation \( y = -124x^2 + 2x \) in the context of a baseball game, we need to analyze the equation.

The given equation represents a downward-opening parabola, as indicated by the negative coefficient of the \( x^2 \) term. The range of a function describes all the possible values of \( y \) (the height of the ball) as \( x \) (the horizontal distance) varies.

1. **Finding the Vertex:** The vertex of the parabola is the highest point (maximum height) the ball reaches during its flight. The vertex can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \), where \( a = -124 \) and \( b = 2 \).

\[
x = -\frac{2}{2(-124)} = \frac{2}{248} = \frac{1}{124}
\]

We substitute this back into the equation to find the corresponding \( y \)-value (the maximum height):

\[
y = -124\left(\frac{1}{124}\right)^2 + 2\left(\frac{1}{124}\right)
\]
\[
y = -124\left(\frac{1}{15376}\right) + \frac{2}{124}
\]
\[
y = -\frac{124}{15376} + \frac{2}{124}
\]

To find a common denominator and do the calculation, let's convert \(\frac{2}{124}\) into a fraction over 15376:

\[
y = -\frac{124}{15376} + \frac{2 \cdot 124}{15376} = -\frac{124}{15376} + \frac{248}{15376} = \frac{124}{15376}
\]

Simplifying \( \frac{124}{15376} \) leads us to:

\[
y = \frac{1}{124}
\]

(Note that the simplification is straightforward, since 124 and 15376 share no further common factors in respect to simple reduction. Hence the maximum height is a little higher than this raw formulation.)

2. **Range Interpretation:** Since the ball is hit upward, it will start at ground level, so the minimum height of the ball (when \( y = 0 \)) occurs when \( x = 0 \), and it rises to its maximum height before eventually falling back to \( y = 0 \) again when it lands.

Thus, we conclude:

- The **minimum height** of the ball is \( y = 0 \) (when the ball is at the ground level).
- The **maximum height** is \( y = \frac{124}{15376} \) (the highest point reached by the ball before descending).

Therefore, the **range** of the ball's height (y-values) is:

\[
0 \leq y \leq \frac{124}{15376}
\]

This means the ball travels from a height of 0 feet (ground level) to a maximum height given by the calculated value. So, in the context of the baseball game, the ball's height varies between these two points as it follows a parabolic path.