Identify the symmetry (if any) in the graphs of the following ...

Question

Identify the symmetry (if any) in the graphs of the following equations.

$y^2-4x^2=4$

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with symmetries. This problem says to determine the type of symmetry for the graph represented by the equation A form multiplied by Y squared minus X squared equal to 16. We're given four possible choices. As our answers. For choice A, we have symmetric about the X axis only choice B symmetric about the Y axis only for choice C symmetric about the X axis Y axis and the origin. And for choice D no symmetry. So we need to test our equation or symmetries about the X axis Y axis and the origin. And so we're gonna start off by looking at the symmetry about the Y axis. And to recall to test for a symmetry about the Y axis to have that symmetry every point X comma Y on the graph. Let's also have the point of minus X comma Y on the graph as well. So that means that we can set X equal to minus X and plug that into our equation. And if we recover the original equation, then this equation is symmetric about the Y axis. So we're gonna make that substitution. So we'll have four equation or multiplied by Y squared minus the quantity. And here for X, we'll replace it with minus X. So we'll have minus X in quantity squared is equal to 16. However, when I square, the quantity of minus X, I get X squared, so we'll have four, Y squared minus X squared is equal to 16. And so I've recovered my original equation. So that means that the equation here is symmetric about the Y axis. Now, we're gonna do something similar for the X axis symmetry. And here we call that for an equation to be symmetric about the X axis that for every point X comma Y on the graph, we must also have the point of X comma minus Y on the graph as well. So to test for this, we're gonna set Y equal to minus Y and plug that into our equation. And if we recover the original equation, then it is symmetric about the X axis. So we're going to have, when I make the substitution of minus Y for Y in our equation, we'll have four multiplied by the quantity of minus Y in quantity squared minus X squared is equal to 16. However, when I square minus Y I get back Y squared, so we'll have four multiplied by Y squared minus X squared is equal to 16. And here again, I have recovered my original equation. So this equation is symmetric about the X axis. Now, finally we need to check for symmetry about the origin. And in this case, in the equation is symmetric about the origin if for every point X comma Y on the graph, we also include the point of minus X comma minus Y. So here to test for this, we're gonna set X equal to minus X in our equation as well as Y equal to minus Y. And if we recover the original equation, then that equation is symmetric about the origin. So plugging in for both um X and Y with these substitutions will have four multiplied by the quantity of minus Y in quantity squared minus the quantity of minus X in quantity squared is equal to 16. Now, when I take my minus Y and squared, I get back Y squared. So I'll have four multiplied by Y squared minus. And here again, when I take minus X and squared, I get X squared and that's gonna be equal to 16. And again, I have recovered my original equation. So this equation is symmetric about the origin. And so now if I come up to my answer choices and look at the, for the correct answer choice that corresponds to answer choice C, the symmetric about the X axis, Y axis and the origin. So I hope that this has been useful. And I'll see you in the next video.

Identify the symmetry (if any) in the graphs of the following equations.
$y^2-4x^2=4$

Key Concepts

Symmetry in Graphs

Conic Sections

Transformations of Functions

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