Problem 74 - Precalculus

Question

The path of a softball is modeled by \(-12.5(y-7.125)=(x-6.25)^{2}\) where the coordinates \(x\) and \(y\) are measured in feet, with \(x=0\) corresponding to the position from which the ball was thrown. (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory.

Step-by-Step Solution

Answer

The highest point of the trajectory is approximately (6.25, 7.125) and the range (horizontal span) can be estimated by tracing along the x-axis to determine where the parabola crosses the x-axis.

Step 1: Graph the Equation

First, graph the equation \(-12.5(y-7.125)=(x-6.25)^{2}\) by inputting it into a graphing utility. This will produce a graphical representation of the trajectory of the softball, which should be a downward-opening parabola.

Step 2: Identify the Vertex

Next, use the trace feature of the graphing utility to approximate the vertex of the parabola, which corresponds to the highest point of the trajectory. Here, the vertex is given by the point (6.25, 7.125).

Step 3: Determine the Range

Lastly, determine the range (horizontal span) of the trajectory. This is estimated by identifying the x-coordinates of the points where the trajectory starts and ends (crosses the x-axis). In this case, the range can be estimated by using the trace function along the x-axis (y = 0) and determining these points. These will be the x-intercept points.

Key Concepts

Parabolic Trajectory

A parabolic trajectory occurs when an object follows a path shaped like a parabola as it moves through space. In this exercise, the softball's path is governed by the quadratic equation \[-12.5(y-7.125)=(x-6.25)^{2}\]. This equation roots from the general form of a parabola with a vertex at \((h, k)\).In the context of projectile motion, the parabola represents the trajectory of the ball from its launch point until it lands. The minus sign on the coefficient \(-12.5(y-7.125)\) indicates that the parabola opens downwards.

• The opening direction is pivotal in understanding the object's motion.
• It tells us that the object will reach a maximum height before descending.

Comprehending this trajectory helps in sports analysis or physics problems involving projectile motion.

Graphing Utility

A graphing utility is a tool, either a physical graphing calculator or a software application, that aids in visualizing mathematical equations. Plotting the softball’s equation \(-12.5(y-7.125)=(x-6.25)^{2}\) using a graphing utility can illustrate the projectile’s path, offering a visual confirmation of its parabolic shape.
When you use a graphing utility, you can easily view complex functions without doing exhaustive manual calculations. This feature is particularly useful when dealing with quadratic equations, as it quickly shows the trajectory and allows you to investigate key features like intercepts and vertices.

• Facilitates understanding by making abstract functions tangible.
• Tools often include features to investigate specific points such as maxima, minima, and roots.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on a graph. For our downward-opening trajectory of the softball, the vertex \((6.25, 7.125)\) indicates the peak of the softball's path. This vertex shows:

• The exact height reached at the apex.
• Where it occurs along the horizontal \(x\) axis, corresponding to the `6.25` feet from where the ball was thrown.

Understanding the vertex is crucial as it helps us calculate the maximum height that the softball reaches during its flight. By interpreting this point using a graphing utility, students can visually and practically engage with theoretical math concepts.

Parabola

A parabola is a fundamental concept in mathematics and physics, representing a symmetrical curve. In the exercise, the equation \(-12.5(y-7.125)=(x-6.25)^{2}\) models a parabola, specifically one that opens downwards. Parabolas are key in studying projectile motion, such as the path of a thrown softball.

• Key characteristics include a vertex, usually the highest or lowest point, and an axis of symmetry.
• The parabola's shape is dictated by its equation format and coefficients, which determine its opening and steepness.

Recognizing a parabola in real-life contexts helps predict and analyze motion paths in various disciplines, from engineering to sports.

Range of Trajectory

The range of a trajectory in a projectile motion problem refers to how far horizontally the object travels. In the case of the softball, determining the range involves finding where the parabolic trajectory intersects the \(x\)-axis.
This range tells us the reach or span of the projectile horizontally. Utilizing a graphing utility allows for a more precise approximation of the range by visually tracing where the trajectory crosses the \(x\)-axis, marking its start and end points.

• The range is key in applications where distance covered by projectiles must be calculated or estimated, like in sports or planning ballistics.
• The \(x\)-intercepts of the trajectory graph provide direct information on the range.