Identify and sketch the graph of each relation. 3 x 2 + 6 x + 3 y 2 − 12 y = 12 3x^2+6x+3y^2-12y=12 3 x 2 + 6 x + 3 y 2 − 12 y = 12

Hello, today we are going to be determining the graph that the equation represents the equation given to us is four X squared plus 64 X plus four Y squared minus 48 Y is equal to 500. In order to approach this problem, we are going to be using the general conic. The general conic is given to us as A X squared plus cy squared plus DX plus EY plus F is equal to zero. With this general conic. There are four conditions that follow it if the coefficients A and C are the same, this implies the equation represents a circle. If the product of A and C is equal to zero, then the equation represents a parabola. If the coefficients of A and C are not equal to each other and their product is greater than zero, then this conic represents an ellipse. And finally, if the product of the coefficients A of C are less than zero or negative numbers, then the equation represents a hyperbola. So in order to approach this problem, we're going to need to identify our A and C coefficients from the equation. The A coefficient is the coefficient in front of the X squared term. And in our equation, that coefficient is four, so A is equal to four, the C coefficient is the coefficient in front of our Y squared term. And in our equation, the Y square term has a coefficient of four. So C is equal to four. Notice that the values of the A NC coefficient are the same, they are both four. This implies that the equation that is given to us represents a circle. And with that being said, the solution to this problem is going to be C So I hope this video helps you in understanding on how to approach this problem. And I'll go ahead and see you all in the next video.

8. Conic Sections

Circles

College Algebra

8. Conic Sections

Circles

2:20 minutes

Problem 41

Textbook Question

Identify and sketch the graph of each relation.
$3x^2+6x+3y^2-12y=12$

Verified step by step guidance

Start by rewriting the given equation: $3x^2 + 6x + 3y^2 - 12y = 12$. Notice that each term involving $x$ and $y$ can be grouped together.

Factor out the common factor of 3 from each term: $3(x^2 + 2x) + 3(y^2 - 4y) = 12$.

Divide the entire equation by 3 to simplify: $x^2 + 2x + y^2 - 4y = 4$.

Complete the square for the $x$ terms: $x^2 + 2x$. Add and subtract $1$ ($(\frac{2}{2})^2$) to complete the square: $(x + 1)^2 - 1$.

Complete the square for the $y$ terms: $y^2 - 4y$. Add and subtract $4$ ($(\frac{-4}{2})^2$) to complete the square: $(y - 2)^2 - 4$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0. In the given relation, the terms involving x and y suggest a quadratic nature, which can represent conic sections such as parabolas, ellipses, or hyperbolas. Understanding how to manipulate and solve these equations is crucial for graphing their corresponding relations.

Graphing Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The equation provided can be rearranged to identify the type of conic section it represents. Knowing how to sketch these graphs involves recognizing their standard forms and key features, such as vertices, foci, and axes of symmetry, which are essential for accurate representation.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to analyze and graph. This technique is particularly useful for rewriting the given relation in a form that reveals its geometric properties, such as the center and radius of a circle or the vertex of a parabola, facilitating the graphing process.

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