Symmetry Determine whether the graphs of the following equatio...

Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x^{5} - x^{3} - 2$

Accepted Answer

Welcome back, everyone consider the following equation that Y equals X to the seventh minus seven X to the fifth minus nine. Find if the graph would be symmetric about the X axis, the Y axis, the origin or none of them A says the X axis B, the Y axis C the origin and the D says none of these. Now given our equation, how can we tell if it's symmetric under these three conditions? Well, let's let Y OK be equal to F FX. Then our graph would be symmetric about the X axis if F FX is equal to negative FX. In other words, if we make our entire equation negative and we get back the same equation, then it's symmetric about the X axis. It's symmetric about the Y axis if F of X equals F of negative X. In other words, we substitute all the terms in our equation with negative X. And finally, it's symmetric about the origin if F of X is equal to the negative value of F of negative X. So we need to test these three conditions with our equation. And if any one of them are true, then we know how our graph will be symmetric. And if none are true, then we can say none of them. So let's start under the first condition testing if it's symmetric about the X axis now about the X axis, OK. That means we're going to try and find the value of negative F of X. So that means it's going to be the negative value of X to the seventh minus seven X to the fifth minus nine, which is going to be equal to negative X to the seventh plus seven, X to the fifth plus nine. Now, is this equal to our original function? No, this is not equal to F of X. So about the X axis, the graph OK is not symmetric. Next, let's test it about the Y axis. OK? Know about the wire axis. OK. We are testing if F of X equals F of negative X. So let's substitute negative X for all the variables in our equation. So that's negative X to the seventh minus seven multiplied by negative X to the fifth minus nine and no negative X raised to the seventh equals negative X to the seventh, negative X to the fifth equals negative X to the fifth multiplied by negative seven will give us positive seven X to the fifth and we'll have back minus nine. Is that equal to our original function? No, it's not. Therefore, that tells us about the Y axis. The graph OK is not symmetric. Now let's do our final test is it symmetric about the origin? Well, OK, we need to evaluate negative F of negative X. So that's going to be the negative value. OK. Let me put a square bracket here of negative X to the seventh minus seven multiplied by negative X to the fifth minus nine. No, we already know what this works out to F of negative X. So it's gonna be the negative value of negative X to the seventh plus seven X to the fifth minus nine. OK? And now this is gonna be work out to be positive X to the seventh. OK, minus seven X to the fifth plus nine. Now is this value equal to, is this value equal to F FX? No, it is not, it's not equal to F FX. Therefore, the graph is not symmetric about the origin. So we know that this graph is not symmetric about the X axis about the Y axis neither about the origin. Therefore D is the correct answer. None of these. Thanks for watching everyone. I hope this video helped.

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$ƒ (x) = x^{5} - x^{3} - 2$

Key Concepts

Symmetry in Graphs

Odd and Even Functions

Graphing Functions

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