Problem 86 - Precalculus : Building Concepts and Connections

Question

A child kicks a ball a distance of 9 feet. The maximum height of the ball above the ground is 3 feet. If the point at which the child kicks the ball is the origin and the flight of the ball can be approximated by a parabola, find an expression for the quadratic function that models the ball's path. Check your answer by graphing the function.

Step-by-Step Solution

Answer

The equation which models the ball's path is \(y = -0.148(x-4.5)^2 + 3\).

Step 1: Identify the vertex of the parabola

The vertex of the parabola is the maximum point. This is given as the maximum height of the ball, which is 3 feet. The ball reaches this height halfway along its trajectory, so at a horizontal distance of \(\frac{9}{2} = 4.5\) feet. So, the vertex is at (h, k) = (4.5, 3).

Step 2: Substitute the vertex into the vertex form of a quadratic function

Firstly, substitute (4.5, 3) into \(y = a(x-h)^2 + k\). This results in \(y = a(x-4.5)^2 + 3\).

Step 3: Substituting another point to find 'a'

Substitute a point from the ball's trajectory into the equation to solve for 'a'. The ball is kicked from the origin, so use the point (0,0). This yields: \(0 = a(0-4.5)^2 + 3\) which simplifies to \(0 = 20.25a + 3.\)

Step 4: Solve for 'a'

To solve for 'a', subtract 3 from both sides of the equation and then divide by 20.25 to get: \(a = -\frac{3}{20.25} = -0.148.\)

Step 5: Write the quadratic function

Substitute 'a' back into the equation from step 2. The equation of the parabola is thus: \(y = -0.148(x-4.5)^2 + 3\).

Key Concepts

Understanding Parabolic Trajectory

A parabolic trajectory is a smooth, symmetrical path that an object follows when acted upon by gravity and no other external forces. This type of path is typical for objects launched into the air, like a soccer ball being kicked. When discussing quadratic functions in math, a parabolic trajectory can be visualized as a "U"-shaped curve, known as a parabola.
In our example, the ball's path is a parabola, with its movement determined by the initial force (the kick) and gravity's pull. The highest point in this path is called the 'vertex', which represents the maximum height the ball reaches.

• The vertical motion of the ball forms the upward and downward movements typical of a parabola.
• The horizontal motion moves the ball across the playing field.

The path's overall shape and direction can be mathematically modeled using quadratic functions, allowing us to predict positions at any point during its flight.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function provides a convenient way to identify the vertex of the parabola, which is incredibly useful in understanding the path's peak. This form is expressed as:

• \[ y = a(x-h)^2 + k \]

where

• \(a\) determines the width and direction of the parabola,
• (h, k) represents the vertex of the parabola.

This is particularly helpful for this exercise because we were given the maximum height (vertex) of the ball's path. Knowing that the ball reaches its apex (highest point) at 3 feet and the horizontal location of this vertex is 4.5 feet, we input these values directly into our vertex form equation. This form allows us to easily manipulate and explore the properties of the quadratic function by simply adjusting the values of \(a, h,\) and \(k\).

Finding Coefficients in Quadratics

Determining the coefficients in a quadratic function is crucial to fully defining the parabola's equation. In our steps, the process involves plugging in known points from the trajectory into the vertex form equation to solve for \(a\). With the vertex known as (4.5, 3) and initial point (0,0), we replace these into our equation:

• \[0 = a(0-4.5)^2 + 3\]

Upon simplifying, we solve for \(a\):

• Subtract 3 from both sides: \(-3 = 20.25a\).
• Divide by 20.25 to isolate \(a\): \(a = -\frac{3}{20.25} = -0.148\).

This process of finding \(a\) is critical because it tells us the "stretch" or "compression" of the parabola and whether it opens upwards or downwards (negative \(a\) implies opening downwards). By knowing \(a\), we can determine the specific shape of the ball's parabolic path.

Graphing Quadratics

Graphing quadratics is a visual way to understand quadratic functions and interpret their key features, such as the vertex and axis of symmetry. To graph the quadratic function:

• Start by plotting the vertex, which is the highest point here (4.5, 3), indicating the ball's peak height.
• Identify the y-intercept by considering where the parabola crosses the y-axis, here at point (0,0).
• Consider the direction of the parabola. With \(a = -0.148\), your parabola opens downwards, representing how the ball ascends to and descends from its vertex.

Use these features to draw the shape of the parabola. This visual representation makes it clear how the path is curved, assisting in predictions of where the object might land or how it flies through the air. It ties together the algebraic function and its practical trajectory, enabling better understanding and further solutions.