Problem 23 - Algebra 2

Question

FOOTBALL When a ball is thrown or kicked, the path it travels is shaped like a parabola. Suppose a football is kicked from ground level, reaches a maximum height of 25 feet, and hits the ground 100 feet from where it was kicked. Assuming that the ball was kicked at the origin, write an equation of the parabola that models the flight of the ball.

Step-by-Step Solution

Answer

The equation of the parabola is \( y = -\frac{1}{100}(x - 50)^2 + 25 \).

Step 1: Identify the Vertex of the Parabola

The vertex of the parabola represents the highest point of the football's flight. In the given problem, the vertex is at a height of 25 feet, and since the ball travels symmetrically, it occurs at half the horizontal distance: \[ (h, k) = \left(\frac{100}{2}, 25\right) = (50, 25) \]

Step 2: Formulate the Vertex Form of the Parabola

The equation of a parabola in vertex form is given by \[ y = a(x - h)^2 + k \] Substitute the vertex \((h, k) = (50, 25)\): \[ y = a(x - 50)^2 + 25 \]

Step 3: Use Known Points to Find 'a'

The parabola passes through the origin (0,0) and we can use this point to find \(a\). Substitute \((x, y) = (0, 0)\) into the equation: \[ 0 = a(0 - 50)^2 + 25 \] Simplifying: \[ 0 = 2500a + 25 \] \[ -25 = 2500a \] \[ a = -\frac{25}{2500} = -\frac{1}{100} \]

Step 4: Write the Final Equation of the Parabola

Substitute \(a = -\frac{1}{100}\) into the vertex form equation: \[ y = -\frac{1}{100}(x - 50)^2 + 25 \] This is the equation that models the flight of the football.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open up or down, depending on its mathematical expression. Parabolas are commonly encountered in problems involving projectile motion, like a ball being kicked or thrown. In this exercise, the path of the kicked football forms a parabolic trajectory. The unique shape of a parabola is determined by its equation, which includes specific terms that define its width, direction, and position on the coordinate plane. The key properties of a parabola include its vertex, axis of symmetry, and direction of opening. Understanding these properties helps us describe and graph the motion of objects following a parabolic path.

Vertex Form

The vertex form of a quadratic equation is particularly useful for identifying the key characteristics of a parabola. It is expressed as \[ y = a(x - h)^2 + k \] where

• \(a\) determines the width and direction of the parabola (upward if positive, downward if negative),
• \(h\) and \(k\) are the coordinates of the vertex, the highest or lowest point of the parabola.

In this problem, the vertex form highlights the highest point of the football's trajectory, which is crucial for understanding the ball's flight pattern. Knowing the vertex helps in easily visualizing the parabola's position on the graph.

Solving for Coefficients

Finding the coefficient \(a\) in the vertex form of a quadratic equation is an essential step. It involves using identifiable points on the parabola. In our given exercise, the known point is the origin (0, 0), where the football was kicked. By substituting these coordinates into the vertex form, we calculated \(a\) as follows:

• Substitute the point into the equation to form \(0 = a(0 - 50)^2 + 25\).
• Simplify to obtain: \(-25 = 2500a\). This leads to solving for \(a = -\frac{1}{100}\).

This coefficient \(a\) reflects how wide and in which direction the parabola opens. In this case, a negative \(a\) value indicates that the parabola opens downward.

Graph of Quadratic Functions

Graphing quadratic functions allows us to visually interpret the behavior of the parabola. A graph provides a clear representation of its shape, position, and direction of opening.

• Start by plotting the vertex, which is ideally the first point to consider as it represents the highest point the football reaches in its trajectory.
• Symmetrically plot additional points along the parabola using the equation, to capture its full shape.
• In this exercise, begin at the origin, which marks the launch point of the parabolic flight, then move to the vertex (50, 25).

Ensure the curve of the graph opens downward, matching the journey of the football as it comes back to the ground, illustrating the entire projectile motion.