Determine the equation of the parabola that opens up, has vertex (−1,−4), and a focal diameter of 28 is a common problem in conic sections that combines understanding of parabola properties, standard forms, and geometric relationships. Whether you're a student working on a math assignment or a math enthusiast exploring conic sections, understanding how to derive a parabola's equation from given parameters is essential. This article will guide you through the step-by-step process to determine the specific parabola that fits these criteria, providing clarity on key concepts like vertex form, focus, directrix, and focal diameter.
Understanding the Basic Properties of Parabolas
Before diving into the calculations, it’s important to grasp the fundamental properties of parabolas, especially those relevant to this problem.
What is a parabola?
A parabola is a symmetric, U-shaped curve that can open upward, downward, left, or right. It is defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.The vertex
The vertex of a parabola is its highest or lowest point (for upward or downward opening parabolas) and serves as a key reference point in its equation.Focus and directrix
The focus is a fixed point inside the parabola, and the directrix is a line outside it. The parabola is the locus of points equidistant from the focus and the directrix.Focal diameter
The focal diameter (also called the latus rectum) is the length of the chord passing through the focus and perpendicular to the axis of symmetry. It measures how wide the parabola is at the level of the focus.Given Parameters and Their Significance
The problem provides three pieces of information:
- Vertex at (−1,−4)
- Parabola opens upward
- Focal diameter of 28
Let's analyze what each of these means for our parabola:
Vertex at (−1,−4)
This point indicates the parabola’s minimum point since it opens upward. The vertex form of a parabola with a vertex at (h, k) is typically written as: \[ y = a(x - h)^2 + k \]Parabola opening upward
This suggests the parabola's axis of symmetry is vertical, and the parabola's focus lies above the vertex.Focal diameter of 28
This length relates to the parabola's width at the focus and is connected to the parameter \( p \) (distance from the vertex to the focus).Understanding how these parameters interrelate is crucial to deriving the equation.
Step-by-Step Process to Find the Equation
To determine the parabola's equation, follow these logical steps:
Step 1: Write the vertex form of the parabola
Since the parabola opens upward with vertex at (−1,−4), the general form is: \[ y = a(x + 1)^2 - 4 \]Our goal is to find the value of \( a \).
Step 2: Connect focal diameter to the parabola's parameters
The focal diameter \( D \) is related to the parameter \( p \) (the distance from vertex to focus) by: \[ D = 4p \] Given \( D = 28 \), then: \[ 4p = 28 \] \[ p = 7 \]This means the focus is 7 units above the vertex, since the parabola opens upward.
Step 3: Determine the focus point
The focus's y-coordinate is: \[ y{focus} = y{vertex} + p = -4 + 7 = 3 \] The focus's x-coordinate remains the same as the vertex's x-coordinate (since the axis of symmetry is vertical): \[ x_{focus} = -1 \] Thus, focus point: \[ F(-1, 3) \]Step 4: Find \( a \) using the focus and the parabola's equation
The focus lies on the parabola, so substitute \( x = -1 \) and \( y = 3 \) into the parabola's equation: \[ y = a(x + 1)^2 - 4 \] \[ 3 = a(0)^2 - 4 \] \[ 3 = -4 \] This does not hold unless \( a \) is undefined, indicating a need to revisit the approach.Instead, recognize that for a parabola opening upward with vertex at (h, k), the standard form can be written as:
\[ (x - h)^2 = 4p(y - k) \]
where \( p \) is the distance from the vertex to the focus (positive for upward opening).
Using this form:
- \( h = -1 \)
- \( k = -4 \)
- \( p = 7 \)
Plug into the standard form:
\[ (x + 1)^2 = 4 \times 7 \times (y + 4) \]
\[ (x + 1)^2 = 28(y + 4) \]
This is the equation of the parabola.
Final Equation and Summary
Based on the above steps, the equation of the parabola that opens upward with a vertex at (−1,−4) and a focal diameter of 28 is:
\[ \boxed{(x + 1)^2 = 28(y + 4)} \]
This form clearly expresses the parabola's geometric properties and provides a straightforward equation for graphing or further analysis.
Additional Insights and Applications
Understanding how to derive the parabola's equation from given parameters has many practical applications:
- Graphing parabolas: Using the vertex form simplifies plotting points and sketching the curve accurately.
- Physics problems: Parabolas describe projectile trajectories where vertex and focus relate to maximum height and focal points.
- Engineering designs: Parabolic shapes are used in satellite dishes and headlights for focusing signals or light.
Summary of Key Concepts
- The vertex form of a parabola opens upward/downward based on the sign of the coefficient.
- The focal diameter relates to the parameter \( p \) via \( D = 4p \).
- The standard form \( (x - h)^2 = 4p(y - k) \) is particularly useful for parabolas opening vertically.
- Coordinate substitution helps verify the focus and ensure the derived equation matches given parameters.
In conclusion, the parabola with vertex (−1,−4), opening upward, and focal diameter 28 is described by the equation:
\[ \boxed{(x + 1)^2 = 28(y + 4)} \]
This comprehensive approach not only solves the problem but also deepens your understanding of parabola properties and equations, enhancing your skills in conic sections analysis.
