Finding The Equation Of A Parabola Vertex At Origin And Passing Through (3,4)

In the realm of conic sections, parabolas hold a special place, characterized by their unique U-shaped curves. These curves, defined mathematically, find applications in diverse fields, from the trajectory of projectiles to the design of satellite dishes. In this comprehensive guide, we will delve into the process of determining the equation of a parabola, focusing on a specific scenario where the vertex is situated at the origin and the axis of symmetry aligns with the x-axis. To solidify our understanding, we will walk through a step-by-step solution for a parabola passing through the point (3, 4). Let's embark on this mathematical journey to unravel the secrets of parabolic equations.

Understanding the Standard Equation of a Parabola

Before we dive into the specifics of our problem, it's crucial to grasp the standard equation of a parabola. When the vertex is at the origin (0, 0) and the axis of symmetry is the x-axis, the parabola's equation takes a simplified form. This form serves as the foundation for our calculations and allows us to express the relationship between the x and y coordinates of any point on the parabola. The standard equation in this scenario is given by:

$$y^2 = 4px$$

Here, 'p' represents a critical parameter – the distance between the vertex and the focus of the parabola. The focus, a fixed point within the parabola, plays a pivotal role in defining its shape and properties. The axis of symmetry acts as a mirror, dividing the parabola into two symmetrical halves. Understanding these fundamental concepts sets the stage for tackling our problem effectively.

Key Elements of the Parabola Equation

To fully comprehend the equation, let's dissect its components. The 'y²' term signifies that the parabola opens either to the right or to the left, depending on the sign of 'p'. If 'p' is positive, the parabola opens to the right, and if 'p' is negative, it opens to the left. The '4p' factor scales the parabola, determining its width. A larger absolute value of 'p' results in a wider parabola, while a smaller value leads to a narrower one. The 'x' term represents the horizontal displacement from the vertex, indicating the position of a point along the x-axis. These elements work in concert to define the parabola's shape, orientation, and position in the coordinate plane.

Applying the Given Conditions

Now, let's turn our attention to the specific problem at hand. We are tasked with finding the equation of a parabola that satisfies three crucial conditions: the vertex is at the origin (0, 0), the axis of symmetry lies along the x-axis, and the parabola passes through the point (3, 4). These conditions act as constraints, guiding us towards the unique equation that describes our parabola. We will leverage the standard equation and the given point to determine the value of 'p', the parameter that dictates the parabola's shape and orientation.

Utilizing the Point (3, 4)

The fact that the parabola passes through the point (3, 4) provides us with a direct connection between the x and y coordinates. This point must satisfy the equation of the parabola, meaning that when we substitute x = 3 and y = 4 into the equation, the equality must hold true. This substitution will allow us to solve for the unknown parameter 'p', effectively unlocking the specific equation of our parabola. It's like having a puzzle piece that fits perfectly into the equation, completing the picture and revealing the parabola's identity.

Solving for 'p'

To find the value of 'p', we will substitute the coordinates of the given point (3, 4) into the standard equation of the parabola, y² = 4px. This process involves replacing 'y' with 4 and 'x' with 3, creating an equation with 'p' as the sole unknown. By isolating 'p' on one side of the equation, we can determine its numerical value, which is essential for defining the parabola's shape and orientation. This step is the heart of our solution, transforming the problem from an abstract concept into a concrete equation.

Substitution and Simplification

Substituting y = 4 and x = 3 into the equation y² = 4px, we get:

$$4^2 = 4p(3)$$

Simplifying this equation, we have:

$$16 = 12p$$

Now, to isolate 'p', we divide both sides of the equation by 12:

$$p = \frac{16}{12} = \frac{4}{3}$$

Thus, we have successfully determined the value of 'p', which is 4/3. This value represents the distance between the vertex and the focus of our parabola and is crucial for defining its specific characteristics.

Writing the Equation of the Parabola

With the value of 'p' in hand, we can now write the complete equation of the parabola. We simply substitute p = 4/3 back into the standard equation y² = 4px. This substitution completes the equation, providing us with a mathematical description of the parabola that satisfies all the given conditions. The resulting equation encapsulates the parabola's shape, orientation, and position in the coordinate plane, allowing us to analyze its properties and make predictions about its behavior.

Final Equation

Substituting p = 4/3 into the equation y² = 4px, we obtain:

$$y^2 = 4(\frac{4}{3})x$$

Simplifying, we get the final equation of the parabola:

$$y^2 = \frac{16}{3}x$$

This equation represents the parabola with its vertex at the origin, axis of symmetry along the x-axis, and passing through the point (3, 4). It's a concise mathematical statement that captures the essence of this specific parabolic curve.

Conclusion

In this exploration, we have successfully navigated the process of finding the equation of a parabola given specific conditions. We began by understanding the standard equation of a parabola with its vertex at the origin and axis of symmetry along the x-axis. We then applied the condition that the parabola passes through the point (3, 4) to solve for the parameter 'p', which governs the parabola's shape and orientation. Finally, we substituted the value of 'p' back into the standard equation to obtain the final equation of the parabola, y² = (16/3)x.

Key Takeaways

This exercise highlights the importance of understanding the standard forms of equations and how to apply given conditions to solve for unknown parameters. The ability to manipulate equations and extract information from them is a fundamental skill in mathematics and physics. By mastering these techniques, we can unlock the secrets of various geometric shapes and physical phenomena, paving the way for deeper understanding and innovation. The journey of finding the equation of a parabola serves as a testament to the power of mathematical reasoning and its ability to describe the world around us.

Further Exploration of Parabolas

While we have successfully found the equation of a specific parabola, the world of parabolas extends far beyond this single example. There are many other variations and applications to explore, each offering unique insights and challenges. Let's delve into some avenues for further exploration:

Different Orientations

Our focus has been on parabolas with their axis of symmetry along the x-axis. However, parabolas can also be oriented with their axis of symmetry along the y-axis. The equation for such parabolas takes a different form, reflecting the change in orientation. Understanding these variations expands our ability to model and analyze parabolic phenomena in diverse contexts.

Vertex Not at the Origin

We've considered parabolas with their vertex at the origin, simplifying the equation. However, parabolas can exist anywhere in the coordinate plane, with their vertex at any point (h, k). The equation for these parabolas incorporates the coordinates of the vertex, adding another layer of complexity and flexibility. Mastering these equations allows us to represent a wider range of parabolic shapes and positions.

Applications of Parabolas

Parabolas are not merely abstract mathematical constructs; they have real-world applications in various fields. From the trajectory of projectiles to the design of satellite dishes and the reflectors in car headlights, parabolas play a crucial role in shaping our technology and understanding the physical world. Exploring these applications provides a tangible connection to the mathematical concepts, making them more engaging and relevant.

Conic Sections

Parabolas belong to a family of curves known as conic sections, which also includes circles, ellipses, and hyperbolas. These curves are formed by the intersection of a plane and a cone, each with its unique properties and equations. Studying conic sections provides a broader perspective on geometric shapes and their relationships, fostering a deeper appreciation for the elegance and interconnectedness of mathematics.

By venturing into these areas of further exploration, we can deepen our understanding of parabolas and their role in the mathematical landscape and the world around us. The journey of mathematical discovery is an ongoing one, with each exploration leading to new insights and challenges.