A Fisherman Throws A Lure Into The Air. On The Interval $0 \leq X \leq 3$, The Fishing Lure Follows A Path Where The Expression $-2x^2 + 4x + 6$ Represents The Elevation Of The Fishing Lure, In Feet, At Any Time
Introduction
In the world of mathematics, parabolas are a fundamental concept that can be used to model various real-world phenomena. One such example is the path of a fishing lure thrown into the air. In this article, we will delve into the mathematical representation of the elevation of a fishing lure, given by the expression $-2x^2 + 4x + 6$, where $0 \leq x \leq 3$. We will explore the properties of this parabolic path, including its vertex, axis of symmetry, and the maximum elevation reached by the lure.
The Parabolic Path
The given expression $-2x^2 + 4x + 6$ represents the elevation of the fishing lure at any time $x$. To understand the shape of this parabola, we need to identify its vertex, axis of symmetry, and the direction in which it opens.
Vertex and Axis of Symmetry
The vertex of a parabola is the point where the parabola changes direction, and it is also the minimum or maximum point of the parabola. To find the vertex, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic expression.
In this case, $a = -2$ and $b = 4$. Plugging these values into the formula, we get:
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging $x = 1$ into the original expression:
Therefore, the vertex of the parabola is at the point $(1, 8)$.
The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at $(1, 8)$, the axis of symmetry is the line $x = 1$.
Direction of the Parabola
The direction of the parabola is determined by the sign of the coefficient $a$. In this case, $a = -2$, which is negative. This means that the parabola opens downward, indicating that the elevation of the fishing lure decreases as it moves away from the vertex.
Maximum Elevation
Since the parabola opens downward, the maximum elevation of the fishing lure occurs at the vertex, which is at the point $(1, 8)$. This means that the maximum elevation of the fishing lure is 8 feet.
Elevation at Specific Points
To find the elevation of the fishing lure at specific points, we can plug the values of $x$ into the original expression. For example, if we want to find the elevation at $x = 0$, we get:
Similarly, if we want to find the elevation at $x = 3$, we get:
Therefore, the elevation of the fishing lure at $x = 0$ is 6 feet, and at $x = 3$ is 0 feet.
Conclusion
In conclusion, the expression $-2x^2 + 4x + 6$ represents the elevation of a fishing lure thrown into the air, where $0 \leq x \leq 3$. We have identified the vertex, axis of symmetry, and the direction of the parabola, and found the maximum elevation of the fishing lure. We have also calculated the elevation of the fishing lure at specific points. This mathematical representation of the elevation of a fishing lure can be used to model various real-world phenomena and can be applied in various fields such as physics, engineering, and computer science.
Applications of Parabolic Paths
Parabolic paths have numerous applications in various fields, including:
- Physics: Parabolic paths are used to model the motion of objects under the influence of gravity, such as the trajectory of a projectile.
- Engineering: Parabolic paths are used to design curves for roads, bridges, and other infrastructure.
- Computer Science: Parabolic paths are used in computer graphics to create realistic animations and simulations.
Real-World Examples
Parabolic paths can be seen in various real-world examples, including:
- Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, follows a parabolic path.
- Road Design: The curves of roads are designed to follow a parabolic path to ensure smooth and safe driving.
- Bridge Design: The curves of bridges are designed to follow a parabolic path to ensure stability and safety.
Conclusion
Q&A: Frequently Asked Questions
Q: What is the vertex of the parabola?
A: The vertex of the parabola is the point where the parabola changes direction, and it is also the minimum or maximum point of the parabola. In this case, the vertex is at the point (1, 8).
Q: What is the axis of symmetry?
A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line x = 1.
Q: What is the direction of the parabola?
A: The direction of the parabola is determined by the sign of the coefficient a. In this case, a = -2, which is negative. This means that the parabola opens downward, indicating that the elevation of the fishing lure decreases as it moves away from the vertex.
Q: What is the maximum elevation of the fishing lure?
A: Since the parabola opens downward, the maximum elevation of the fishing lure occurs at the vertex, which is at the point (1, 8). This means that the maximum elevation of the fishing lure is 8 feet.
Q: How do I find the elevation of the fishing lure at specific points?
A: To find the elevation of the fishing lure at specific points, you can plug the values of x into the original expression. For example, if you want to find the elevation at x = 0, you get:
y = -2(0)^2 + 4(0) + 6 = 6
Similarly, if you want to find the elevation at x = 3, you get:
y = -2(3)^2 + 4(3) + 6 = 0
Q: What are some real-world examples of parabolic paths?
A: Parabolic paths can be seen in various real-world examples, including:
- Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, follows a parabolic path.
- Road Design: The curves of roads are designed to follow a parabolic path to ensure smooth and safe driving.
- Bridge Design: The curves of bridges are designed to follow a parabolic path to ensure stability and safety.
Q: How can I apply parabolic paths in my field of study?
A: Parabolic paths have numerous applications in various fields, including physics, engineering, and computer science. You can apply parabolic paths in your field of study by:
- Modeling real-world phenomena: Use parabolic paths to model the motion of objects under the influence of gravity, such as the trajectory of a projectile.
- Designing curves: Use parabolic paths to design curves for roads, bridges, and other infrastructure.
- Creating realistic animations: Use parabolic paths in computer graphics to create realistic animations and simulations.
Q: What are some common mistakes to avoid when working with parabolic paths?
A: Some common mistakes to avoid when working with parabolic paths include:
- Not identifying the vertex: Make sure to identify the vertex of the parabola to determine the direction and maximum elevation.
- Not using the correct formula: Use the correct formula to find the elevation of the fishing lure at specific points.
- Not considering the axis of symmetry: Make sure to consider the axis of symmetry when designing curves or modeling real-world phenomena.
Conclusion
In conclusion, parabolic paths have numerous applications in various fields, including physics, engineering, and computer science. By understanding the properties of parabolic paths, you can apply them in your field of study to model real-world phenomena, design curves, and create realistic animations. Remember to avoid common mistakes and use the correct formula to find the elevation of the fishing lure at specific points.