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Magazine Chapter 8: Problem 10
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y^{2}+12 x-6 y-51=0\)
Short Answer
Expert verified
Yes, it is a parabola. Standard form: \((y - 3)^2 = -12(x - 5)\).
Step by step solution
01
Identify the Type of Equation
Identify the form of the given equation. The equation provided is: \(y^{2} + 12x - 6y - 51 = 0\). We notice that this equation is a quadratic because of the \(y^2\) term. A parabola in standard form typically involves one squared term, either \(x^2\) or \(y^2\). Hence, this equation could represent a parabola.
02
Rearrange the Equation
Group the terms involving \(y\) together: \(y^2 - 6y + 12x - 51 = 0\). To rewrite in standard form, we aim to complete the square for the \(y\) terms.
03
Complete the Square for the y Terms
To complete the square for \(y^2 - 6y\), take half of the coefficient of \(y\) which is \(-6\), giving \(-3\), and square it to get \(9\). Add and subtract \(9\) to the equation: \(y^2 - 6y + 9 + 12x - 51 - 9 = 0\).
04
Simplify the Equation
The equation now is \((y - 3)^2 + 12x - 60 = 0\). Simplify this to \((y - 3)^2 = -12x + 60\).
05
Express in Standard Form
Rewrite the equation into the standard form of a parabola, which looks like \((y - k)^2 = 4p(x - h)\). Here, move all terms except \((y - 3)^2\) to one side: \((y - 3)^2 = -12(x - 5)\).
06
Identify Parameters and Confirm
Compare \((y - 3)^2 = -12(x - 5)\) with \((y - k)^2 = 4p(x - h)\). Here, the vertex is at \((h, k) = (5, 3)\) and \(4p = -12\), giving \(p = -3\). This equation confirms it is a parabola that opens to the left.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
In the world of algebra, quadratic equations hold a special place. They are expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
These equations can appear in different forms and reveal a variety of solutions. Typically, they include a squared term, making them non-linear.
Importantly, not all quadratic equations strictly involve \(x\); some, as in our case, involve \(y^2\).
These equations can appear in different forms and reveal a variety of solutions. Typically, they include a squared term, making them non-linear.
Importantly, not all quadratic equations strictly involve \(x\); some, as in our case, involve \(y^2\).
- If a squared term is \(x^2\), it describes vertical parabolas.
- If it is \(y^2\), it describes horizontal parabolas.
Completing the Square
Completing the square is a technique used to transform a quadratic equation into a form that is easier to work with.
In our exercise, we needed to arrange the equation \(y^2 - 6y + 12x - 51 = 0\) to complete the square for \(y\).
Here's how it works:
In our exercise, we needed to arrange the equation \(y^2 - 6y + 12x - 51 = 0\) to complete the square for \(y\).
Here's how it works:
- Take the coefficient of \(y\), which is \(-6\) in this case.
- Halve it to get \(-3\), then square \(-3\) to get \(9\).
- Add and subtract this square inside the equation: \(y^2 - 6y + 9 - 9 = 0\).
Standard Form of a Parabola
The standard form of a parabola helps us quickly identify key features like its vertex and the direction it opens.
Typically, for parabolas opening horizontally, the standard form is \((y-k)^2 = 4p(x-h)\). In our case, the rearranged equation \((y - 3)^2 = -12(x - 5)\) fits this model.
The key components in this form include:
Typically, for parabolas opening horizontally, the standard form is \((y-k)^2 = 4p(x-h)\). In our case, the rearranged equation \((y - 3)^2 = -12(x - 5)\) fits this model.
The key components in this form include:
- \(h\) and \(k\), which specify the vertex of the parabola.
- \(4p\), which helps determine the parabola's direction and focal width.
Vertex of a Parabola
The vertex of a parabola is a point of utmost importance. It serves as the peak or the lowest point of the curve, depending on its orientation.
In standard form \((y-k)^2 = 4p(x-h)\), the vertex is located at \(h, k\). For our equation \((y - 3)^2 = -12(x - 5)\), the vertex is at \(5, 3\).
The vertex not only indicates a pivotal point in the parabola's path but also symmetrically divides the parabola.
In standard form \((y-k)^2 = 4p(x-h)\), the vertex is located at \(h, k\). For our equation \((y - 3)^2 = -12(x - 5)\), the vertex is at \(5, 3\).
The vertex not only indicates a pivotal point in the parabola's path but also symmetrically divides the parabola.
- It helps in understanding the graph's layout.
- It provides a quick reference as to how the parabola behaves.
