In Exercises 67–68, graph each semiellipse. y = √16 - 4x²

Question

Accepted Answer

Hello, today we're gonna be identifying the graph of the semi ellipse given. Now, the equation of the semi ellipse is given to us as Y is equal to one third times the square root of the quantity nine minus X squared. Now the best way to identify the graph of the semi ellipse is to represent the current equation of the graph as a regular ellipse. In order to do that, we want to go ahead and set this equation equal to one. And the first step to set the equation equal to one is to multiply both sides of the equation by three. So that way we can get rid of the fraction here, multiplying both sides by three is going to give us three Y is equal to the square root of nine minus X squared. Next, we want to get rid of the square root on the right hand side. And in order to do that, we can square both sides of the equation. Three Y squared is going to give us nine Y squared and that is going to equal to nine minus X squared. Next, we want to add X squared to both sides of the equation that is going to give us X squared plus nine, Y squared is equal to nine. And finally, in order to set the equation equal to one, we want to divide both sides of the equation by nine. That is going to give us X squared over nine plus Y squared over one is equal to one. Now, there's one thing to note about this equation of the ellipse. The value underneath the X value is greater than the value underneath the Y value. That means that the major axis of the ellipse is going to be the X axis. So what that means is that we're going to have the shape of an ellipse where the major vertices are going to be along the X axis and the minor vertices are going to be along the Y axis. What we wanna do now is we want to solve for the values of the major and minor vertices. We're going to represent the values of the major vertices with the variable A and we're going to represent the values of the minor vertices with the variable B. So we're going to allow A squared to equal to the value underneath X squared which is currently nine. And we're going to allow B squared to equal to the value under Y squared, which is one solving for A on the left hand side by taking the square root of both sides is going to give us A is equal to plus or minus three. And taking the square root of both sides of the B equation is going to give us B is equal to plus or minus one. What that means is that the coordinates of the major minor vertices or at least the coordinates of the major vertices are going to be three comma zero and negative three comma zero. And the coordinates of the minor vertices are going to be zero, comma one and zero, comma negative one. Now keep in mind that the graph given to us is a semi ellipse. So even though we have the shape of the ellipse is the shape of the ellipse going to cover the upper half of the ellipse or the lower half of the ellipse. Well, in order to verify this, we can go ahead and solve for the Y intercept. Since the ma the minor vertices lie on the Y axis. In order to solve for the Y intercept, we can take the original equation and set X is equal to zero. Then solve for Y that is going to give us Y is equal to one third times the square root of nine minus zero, which is going to give us nine. So the square root of nine is going to simplify to give us three. So Y is equal to one third times three and one third times three is going to simplify to give us positive one. So that means that the position of the minor axis of the semi ellipse is going to be at one comma zero. And that verifies that the graph of the semi ellipse is going to have the major vertices at negative three comma zero and positive three comma zero and the minor vertices at zero comma one. And the option that accurately represents the semi ellipse is going to be option A. So A is going to be the answer to this question. And with that being said, I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 67–68, graph each semiellipse. y = √16 - 4x²

Key Concepts

Semiellipse

Graphing Quadratic Functions

Intercepts and Symmetry

Watch next