find an x^' y^'-equation for the graph of the given x y-equation so that the center or, in the c

step 1 Welcome to this lesson. In this lesson, find the center of this curve, then the vertices of it.

Find an x^' y^'-equation for the graph of the given x...

Key Concepts

Completing the Square

Completing the square is a method used to rewrite a quadratic expression in a way that highlights its square structure. This process is essential for transforming a general quadratic equation into its standard form, which facilitates analyzing and graphing conic sections by making key features such as the center or vertex apparent.

Translation of Axes

Translation of axes involves shifting the coordinate system so that a specific point, such as the center of a conic or the vertex of a parabola, becomes the new origin. This technique simplifies the equation by eliminating linear terms, making it easier to identify the conic's shape, orientation, and other properties.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Each type has a characteristic standard form that reveals its properties. Transforming a conic's equation into this standard form is crucial for determining its classification and understanding its geometric features.

Hyperbola

A hyperbola is a type of conic section that appears when the quadratic equation represents a difference of squares, typically leading to two separate branches. Recognizing a hyperbola involves identifying the differing signs associated with the squared terms in its standard equation form. This recognition allows for determining the location of its center, asymptotes, and overall orientation.

Graphing Conic Sections

Graphing conic sections entails drawing the curve along with its key features such as the vertices, center, foci, and asymptotes (if applicable), as well as both the original and transformed coordinate axes. This practice is crucial for visualizing the properties and behavior of the conic, and it helps in confirming the correctness of algebraic manipulations and transformations.

Transcript

0:00 Welcome to this lesson.

00:02 In this lesson, find the center of this curve, then the vertices of it.

00:09 Then after that, we will draw the whole curve.

00:15 Looking at the nature, we have x squared plus 4 y squared.

00:21 So this can be an ellipse.

00:27 So let's group them.

00:30 I mean the variables, we have 32y.

00:36 This is equal to zero and now let's try to write it in this form x minus x squared a squared plus y minus k squared b squared is equal to one so this is what we want to write the both in us so it may go through various transformations before we can get this time now we have x squared that's four writing the whole equation again now we know this is x minus 0 squared okay then we can factor out four so that we have y squared minus 8 y that is equal to 0 and now there's some other transformation we can also do so that we can have the form y minus k so here for us to get the constant path, we divide the 8 by 2, then we square that.

02:12 So this would give us negative 4 squared, which is 16.

02:22 So we can have y squared minus 8y plus 16, that is called to 0.

02:29 We can write the whole thing as y minus 4 squared.

02:39 To zero but not as early zero because you have added 16 times four to it so here we would have 64 rather okay yeah so that this and that becomes the same and now we would divide through by 64 and this becomes x minus zero squared this is y minus four squared all on 16 and that is equal to one okay so at this point we can compare it to what we started with and here we have the center which is equal to the hk so we have here the center as zero and four okay so you have zero and four 0 and 4 as the center.

04:11 Now the bigger between the a and a b, okay, we have the a as that bigger one.

04:22 It means that the major address would be parallel to the x.

04:29 Now we have the center, which is 04.

04:37 1, 2, 3, 4.

04:40 1, 2, 3, 4.

04:41 1, 2, 3, 4...

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