Graph each of the following equations. 9 y^2=9-x^2 |...

Name: Problem 24, Chapter 13: Elementary and Intermediate Algebra
Uploaded: 2020-06-12T20:48:14-08:00
Duration: 2 min 48 s
Channel: Will Cadelina
Description: Graph each of the following equations. 9 y^2=9-x^2

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Understanding conic sections involves recognizing their algebraic representations and geometric properties, which are critical for graphing and analyzing these curves in the Cartesian plane.

Ellipse

An ellipse is a type of conic section that can be defined as the set of points in a plane where the sum of the distances from two fixed points (foci) is constant. It has a standard form equation that allows identification of its center, vertices, and axes lengths. Graphing an ellipse involves plotting its vertices and using its symmetry about the axes.

Standard Form Transformation

Transforming an equation into its standard form is a key step in analyzing and graphing conic sections. This process involves algebraic manipulation to rewrite the equation so that all terms are clearly structured, which allows for the easy identification of key features like the center, axes lengths, and orientation of the conic.

Graphing in the Cartesian Plane

Graphing equations in the Cartesian plane requires plotting points that satisfy the equation and making use of the symmetry inherent in conic sections. For an ellipse, this involves identifying and plotting the vertices along the major and minor axes, then sketching the smooth, closed curve that best represents the equation.

Transcript

00:01 The graph of our equation, 9 y squared, is equal to 9 minus x squared.

00:06 We have to start by trying to fit this into our standard equation of an ellipse.

00:10 So what we're going to do is just add x squared to both sides to cancel out negative x squared.

00:23 Okay, and now from here, what we want to do is we want to get a 1 on the right -hand side of our equation.

00:28 So what we can do is just divide both sides by 9, and we'll end up with 1 on the right -hand side.

00:46 Okay, that 9 on the bottom of the 9 -y squared is going to cancel out.

00:51 That above 9 so it can be y squared.

00:54 And this is our standard equation of our ellipse because y squared is just equal to y squared over 1.

01:02 Now we can use our information to graph the equation.

01:06 Okay, to find our center, our center is going to be defined at the points h and k.

01:10 And since there are no values underneath the squared symbol with the x and the y, that means that h and k are both equal to zero, so our center is at zero.

01:24 Okay, now to find our a value.

01:26 We can see that a squared is going to be equal to nine.

01:29 That means that a is equal to the square root of nine, which is equal to three.

01:35 We'll do the same thing for our b value...

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