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Magazine Chapter 12: Problem 11
Find the vertex and focus of the parabola. Sketch its graph, showing the focus. $$ y^{2}-4 y-2 x-4=0 $$
Short Answer
Expert verified
The vertex is \((-4, 2)\) and the focus is \((-3.5, 2)\).
Step by step solution
01
Rearrange the equation
Start by rearranging the given equation in standard form. The original equation is \( y^2 - 4y - 2x - 4 = 0 \). Rearrange terms to get \( y^2 - 4y = 2x + 4 \).
02
Complete the square
To express \( y^2 - 4y \) in a perfect square form, take half of the coefficient of \( y \), which is \(-4\), and square it. \((-4/2)^2 = 4\). Add and subtract 4 to the left-hand side: \( (y^2 - 4y + 4) - 4 = 2x + 4 \). This simplifies to \( (y - 2)^2 = 2x + 8 \).
03
Rewrite the equation in vertex form
The equation \( (y - 2)^2 = 2(x + 4) \) is now in the standard form of a parabola, \( (y - k)^2 = 4p(x - h) \), where \((h,k)\) is the vertex and \(p\) indicates the distance to the focus.
04
Identify the vertex
From the equation \( (y - 2)^2 = 2(x + 4) \), identify the vertex as \((-4, 2)\) by comparing it to the standard form \((y - k)^2 = 4p(x - h)\), which is \((y - 2)^2 = 2(x - (-4))\).
05
Calculate the distance to the focus
The term \(2\) corresponds to \(4p\), giving \(p = \frac{2}{4} = \frac{1}{2}\). This value is the distance from the vertex to the focus.
06
Determine the focus
Since the parabola opens to the right, the focus \((h + p, k)\) is \((-4 + \frac{1}{2}, 2) = (-3.5, 2)\).
07
Sketch the graph
Plot the vertex at \((-4, 2)\) and the focus at \((-3.5, 2)\). The parabola opens to the right, indicating that for every increasing positive interval of \(x\), \(y\) moves above and below \(2\) such that \((y - 2)^2 = 2(x + 4)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertex
The vertex of a parabola is a crucial point that indicates the peak or the lowest point, depending on the parabola's orientation. It serves as a critical tool in graphing parabolas since it defines the axis of symmetry. For any parabola of the form
Always start the sketch with the vertex; it anchors the curve in the coordinate plane.
- For a standard form like \((y - k)^2 = 4p(x - h)\), the vertex is found at the coordinates \((h, k)\).
- In our problem with the equation \((y - 2)^2 = 2(x + 4)\), the vertex is directly obtained by inspecting the equation, which is \((-4, 2)\).
Always start the sketch with the vertex; it anchors the curve in the coordinate plane.
Focus
The focus of a parabola is an intrinsic point that, along with the directrix, defines a parabola as the set of points equidistant to the focus and directrix. Understanding its location is essential for graphing and analyzing the parabola.
From the standard form equation \((y - k)^2 = 4p(x - h)\),
From the standard form equation \((y - k)^2 = 4p(x - h)\),
- The focus is positioned a distance \(p\) from the vertex, along the axis of symmetry.
- Given \((y - 2)^2 = 2(x + 4)\), \(4p = 2\) suggesting \(p = \frac{1}{2}\).
- The focus is therefore \((-4 + \frac{1}{2}, 2)\), or \((-3.5, 2)\).
Completing the Square
Completing the square is a method used to transform quadratic equations into a more manageable form, often a vertex form, for easier graphing and identifying key features like the vertex.
- In the problem, we start by taking the quadratic component \(y^2 - 4y\).
- To complete the square, take half of \(-4\), square it to get \(4\), and add/subtract this term to form \((y - 2)^2\).
- This converts the equation into \((y - 2)^2 = 2(x + 4)\), a readable format that immediately hints at the vertex and focus.
Parabola Graph
Graphing a parabola involves plotting the vertex and the focus, then sketching the curve based on these points and the standard form of the equation.
- Start with the vertex at \((-4, 2)\). This is the parabola's starting point.
- Next, place the focus at \((-3.5, 2)\). Being on the same level as the vertex implies the parabola opens horizontally.
- The axis of symmetry for this specific case is vertical through \(y=2\).
- Knowing how it opens, use the focus to guide the sketch to the right from the vertex.
Equation Transformation
Transforming a quadratic equation into different forms can reveal critical properties of the parabola and simplify the graphing process. The standard form helps to directly extract the vertex and focus positions.
- The transformation usually begins with rearranging, as seen: \(y^2 - 4y = 2x + 4\).
- After completing the square to become \((y - 2)^2 = 2(x + 4)\), the equation presents useful elements for analysis and graphing.
- It solves the challenge of pinpointing both the vertex and focus without requiring graphing initially.
