Graphing the Height Function

Oct 13, 2025 by magicalceylontours.com 43 views

Introduction

In this article, we will explore the height of Stefano's sunglasses as they fall from the edge of a canyon. The height of the sunglasses is given by the function ${$ h(t) $}$, where ${$ t $}$ is the time in seconds. We will analyze the given table and use mathematical concepts to understand the behavior of the sunglasses' height over time.

The Height Function

The height function ${$ h(t) $}$ is a mathematical representation of the height of the sunglasses at any given time ${$ t $}$. The function is given by the equation:

${$ h(t) = -5t^2 + 50t + 100 $}$

where ${$ t $}$ is the time in seconds.

Analyzing the Table

The table below shows the height of the sunglasses at different times:

Time (s)	Height (m)
0	100
2	90
4	80
6	70
8	60
10	50

Calculating the Height

We can use the height function to calculate the height of the sunglasses at any given time. For example, let's calculate the height of the sunglasses at ${$ t = 5 $}$ seconds.

${$ h(5) = -5(5)^2 + 50(5) + 100 $}$

${$ h(5) = -125 + 250 + 100 $}$

${$ h(5) = 225 $}$

Therefore, the height of the sunglasses at ${$ t = 5 $}$ seconds is ${$ 225 $}$ meters.

Graphing the Height Function

We can graph the height function to visualize the behavior of the sunglasses' height over time. The graph will show a parabola that opens downwards, indicating that the height of the sunglasses decreases over time.

Key Features of the Graph

The graph of the height function has several key features:

Vertex: The vertex of the parabola is the point where the height of the sunglasses is maximum. In this case, the vertex is at ${$ t = -10 $}$ seconds, which is not shown in the table.
Axis of Symmetry: The axis of symmetry of the parabola is the vertical line that passes through the vertex. In this case, the axis of symmetry is the line ${$ x = -10 $}$.
Intercepts: The intercepts of the parabola are the points where the height of the sunglasses is zero. In this case, the intercepts are at ${$ t = 0 $}$ and ${$ t = 20 $}$.

Interpreting the Graph

The graph of the height function shows that the height of the sunglasses decreases over time. The height of the sunglasses is maximum at the vertex, and it decreases as we move away from the vertex. The graph also shows that the height of the sunglasses is zero at the intercepts.

Conclusion

In this article, we analyzed the height of Stefano's sunglasses as they fall from the edge of a canyon. We used the height function to calculate the height of the sunglasses at different times and graphed the height function to visualize the behavior of the sunglasses' height over time. The graph showed that the height of the sunglasses decreases over time, and it is maximum at the vertex.

Mathematical Concepts

This article used several mathematical concepts, including:

Functions: A function is a mathematical representation of a relationship between variables. In this case, the height function ${$ h(t) $}$ represents the relationship between the time ${$ t $}$ and the height of the sunglasses.
Graphs: A graph is a visual representation of a function. In this case, the graph of the height function shows the behavior of the sunglasses' height over time.
Parabolas: A parabola is a type of curve that is symmetrical about a vertical line. In this case, the graph of the height function is a parabola that opens downwards.

Real-World Applications

This article has several real-world applications, including:

Physics: The height function can be used to model the motion of objects under the influence of gravity.
Engineering: The height function can be used to design and optimize the trajectory of projectiles.
Computer Science: The height function can be used to model and analyze the behavior of algorithms.

Future Research Directions

This article has several future research directions, including:

Modeling more complex motion: The height function can be used to model more complex motion, such as the motion of objects under the influence of multiple forces.
Analyzing the behavior of algorithms: The height function can be used to analyze the behavior of algorithms and optimize their performance.
Developing new mathematical models: The height function can be used to develop new mathematical models that can be used to analyze and optimize complex systems.
Q&A: The Height of Stefano's Sunglasses =============================================

Q: What is the height function?

A: The height function ${$ h(t) $}$ is a mathematical representation of the height of the sunglasses at any given time ${$ t $}$. It is given by the equation ${$ h(t) = -5t^2 + 50t + 100 $}$.

Q: How do I calculate the height of the sunglasses at a given time?

A: To calculate the height of the sunglasses at a given time, you can plug the time into the height function. For example, to calculate the height of the sunglasses at ${$ t = 5 $}$ seconds, you would use the equation ${$ h(5) = -5(5)^2 + 50(5) + 100 $}$.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point where the height of the sunglasses is maximum. In this case, the vertex is at ${$ t = -10 $}$ seconds.

Q: What is the axis of symmetry?

A: The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is the line ${$ x = -10 $}$.

Q: What are the intercepts of the parabola?

A: The intercepts of the parabola are the points where the height of the sunglasses is zero. In this case, the intercepts are at ${$ t = 0 $}$ and ${$ t = 20 $}$.

Q: How does the graph of the height function show the behavior of the sunglasses' height over time?

A: The graph of the height function shows that the height of the sunglasses decreases over time. The height of the sunglasses is maximum at the vertex, and it decreases as we move away from the vertex.

Q: What are some real-world applications of the height function?

A: The height function has several real-world applications, including:

Physics: The height function can be used to model the motion of objects under the influence of gravity.
Engineering: The height function can be used to design and optimize the trajectory of projectiles.
Computer Science: The height function can be used to model and analyze the behavior of algorithms.

Q: What are some future research directions for the height function?

A: Some future research directions for the height function include:

Modeling more complex motion: The height function can be used to model more complex motion, such as the motion of objects under the influence of multiple forces.
Analyzing the behavior of algorithms: The height function can be used to analyze the behavior of algorithms and optimize their performance.
Developing new mathematical models: The height function can be used to develop new mathematical models that can be used to analyze and optimize complex systems.

Q: How can I use the height function in my own research or projects?

A: You can use the height function in your own research or projects by applying it to a specific problem or scenario. For example, you could use the height function to model the motion of a projectile or to analyze the behavior of an algorithm.

Q: What are some common mistakes to avoid when working with the height function?

A: Some common mistakes to avoid when working with the height function include:

Not checking the domain of the function: Make sure to check the domain of the function to ensure that it is valid for the given input.
Not checking the range of the function: Make sure to check the range of the function to ensure that it is valid for the given output.
Not considering the units of the function: Make sure to consider the units of the function to ensure that it is accurate and consistent.

Q: How can I learn more about the height function and its applications?

A: You can learn more about the height function and its applications by:

Reading books and articles: Read books and articles on the topic of the height function and its applications.
Taking online courses: Take online courses on the topic of the height function and its applications.
Joining online communities: Join online communities related to the topic of the height function and its applications.