Finding The Standard Equation Of A Parabola With Vertical Axis
In mathematics, a parabola is a U-shaped curve that can be defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Parabolas are fundamental shapes in mathematics and physics, appearing in various applications, from the trajectory of projectiles to the design of satellite dishes. One of the key aspects of understanding parabolas is the ability to determine their equations based on given conditions. This article focuses on finding the standard equation of a parabola that has a vertical axis, given its vertex and an x-intercept. We will walk through the process step-by-step, providing a comprehensive guide for students and enthusiasts alike.
Understanding the Standard Equation of a Parabola with a Vertical Axis
Before diving into the problem-solving process, it is crucial to understand the standard equation of a parabola with a vertical axis. The standard form of the equation is:
(x - h)² = 4p(y - k)
Where:
- (h, k) represents the coordinates of the vertex of the parabola.
- p is the distance between the vertex and the focus, and also the distance between the vertex and the directrix. The sign of p determines the direction in which the parabola opens: if p > 0, the parabola opens upwards; if p < 0, the parabola opens downwards.
This equation provides a concise way to describe a parabola's shape and position in the coordinate plane. The vertex (h, k) serves as the central point around which the parabola is symmetric, and the parameter p dictates the parabola's curvature and orientation. By understanding these parameters, we can effectively determine the equation of a parabola given specific conditions.
Problem Statement: Finding the Equation Given Vertex and x-intercept
Now, let's tackle the specific problem at hand. We are tasked with finding the standard equation of a parabola that has a vertical axis and satisfies the following conditions:
- Vertex: (3, -3)
- x-intercept: -2
The vertex provides us with the values of h and k, which are essential components of the standard equation. The x-intercept gives us a point on the parabola, which we can use to solve for the remaining unknown, p. By systematically using this information, we can arrive at the desired equation.
Step-by-Step Solution
To find the standard equation of the parabola, we'll follow these steps:
1. Substitute the Vertex Coordinates
We know the vertex is (3, -3), so we can substitute h = 3 and k = -3 into the standard equation:
(x - 3)² = 4p(y - (-3))
Simplifying, we get:
(x - 3)² = 4p(y + 3)
2. Use the x-intercept to Find p
The x-intercept is the point where the parabola crosses the x-axis, meaning the y-coordinate is 0. We are given the x-intercept as -2, so the point is (-2, 0). We can substitute these coordinates into the equation we derived in step 1:
(-2 - 3)² = 4p(0 + 3)
This simplifies to:
(-5)² = 12p
25 = 12p
Now, we can solve for p:
p = 25 / 12
3. Write the Standard Equation
Now that we have the value of p, we can substitute it back into the equation from step 1:
(x - 3)² = 4(25 / 12)(y + 3)
Simplifying further:
(x - 3)² = (25 / 3)(y + 3)
This is the standard equation of the parabola that satisfies the given conditions. It describes a parabola with a vertex at (3, -3) that opens upwards, as p is positive, and passes through the x-intercept -2.
Alternative Forms of the Equation
While the standard equation is useful for identifying the vertex and the value of p, the equation can also be expressed in other forms. One common form is the general form of a quadratic equation:
Ax² + Bx + Cy² + Dy + E = 0
To convert our standard equation to the general form, we need to expand and rearrange the terms:
(x - 3)² = (25 / 3)(y + 3)
Expand the left side:
x² - 6x + 9 = (25 / 3)(y + 3)
Multiply both sides by 3 to eliminate the fraction:
3(x² - 6x + 9) = 25(y + 3)
3x² - 18x + 27 = 25y + 75
Rearrange the terms to get the general form:
3x² - 18x - 25y - 48 = 0
This is the general form of the equation of the parabola. While it doesn't directly reveal the vertex or the value of p, it is a valid representation of the same parabola. Understanding how to convert between different forms of the equation is a valuable skill in algebra and calculus.
Graphing the Parabola
To visualize the parabola, we can graph it using the standard equation or the general form. Graphing calculators or online tools can be used to plot the equation. Alternatively, we can use the information we have gathered to sketch the graph manually.
We know the vertex is at (3, -3), and the parabola opens upwards because p is positive. We also know that the parabola passes through the x-intercept (-2, 0). To get a more accurate graph, we can find additional points by substituting different values of x into the standard equation and solving for y. For example, we can find the y-intercept by setting x = 0:
(0 - 3)² = (25 / 3)(y + 3)
9 = (25 / 3)(y + 3)
Multiply both sides by 3 / 25:
27 / 25 = y + 3
y = 27 / 25 - 3
y = -48 / 25
So, the y-intercept is approximately (0, -1.92). With the vertex, x-intercept, and y-intercept, we can sketch a reasonably accurate graph of the parabola. The graph will show a U-shaped curve opening upwards, with the vertex at the lowest point and passing through the given intercepts.
Key Concepts and Applications
Understanding the standard equation of a parabola and how to derive it from given conditions is essential in various areas of mathematics and physics. Here are some key concepts and applications:
- Conic Sections: Parabolas are one of the four conic sections, which are curves formed by the intersection of a plane and a double cone. The other conic sections are circles, ellipses, and hyperbolas. The properties and equations of conic sections are fundamental in analytic geometry.
- Projectile Motion: The trajectory of a projectile, such as a ball thrown into the air, is often modeled as a parabola. The standard equation can be used to analyze the projectile's path, including its maximum height and range.
- Reflective Properties: Parabolas have a unique reflective property: any ray of light or sound that enters the parabola parallel to its axis of symmetry will be reflected to the focus. This property is used in the design of satellite dishes, telescopes, and spotlights.
- Optimization Problems: Parabolas can be used to model optimization problems, such as finding the maximum or minimum value of a quadratic function. The vertex of the parabola represents the optimal point.
- Engineering and Architecture: Parabolas are used in the design of bridges, arches, and other structures. Their shape provides structural strength and aesthetic appeal.
By mastering the concepts and techniques discussed in this article, you will gain a solid foundation for working with parabolas and their applications.
Conclusion
In this article, we have explored the process of finding the standard equation of a parabola with a vertical axis, given its vertex and an x-intercept. We started by understanding the standard equation and its parameters, then systematically substituted the given information to solve for the unknown parameter p. We also discussed alternative forms of the equation, how to graph the parabola, and key concepts and applications related to parabolas. This comprehensive guide provides a solid foundation for understanding and working with parabolas in various mathematical and real-world contexts. Remember to practice these steps with different examples to solidify your understanding. Mastering the standard equation will help you in a variety of mathematical applications. The standard equation gives a clear representation of the parabola's key features. Understanding the relationship between the standard equation and the parabola's graph can be valuable for problem-solving. Keep in mind that the standard equation is not the only form, but it is often the most useful for identifying key parameters. The ability to derive and interpret the standard equation is essential for further studies in mathematics and related fields. The standard equation is a building block for more advanced concepts. The standard equation is a powerful tool in your mathematical toolkit. Don't hesitate to review this process and practice more examples to become fully proficient in finding the standard equation of a parabola. With consistent effort, you will be able to confidently tackle any problem involving parabolas.
