Symmetry Determine whether the graphs of the ...

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.

$ƒ (x) = x^{4} + 5 x^{2} - 12$

Accepted Answer

Welcome back, everyone. Consider the following equation Y equals X to the sixth plus seven X squared minus nine. Find if the graph would be symmetric about the X axis, the Y axis, the origin or about none of them. A says it would be symmetric about the X axis B symmetric about the Y axis C symmetric about the origin or D says none of these. Now, given our equation, how can we figure out if the graph would be symmetric under each of these conditions? Well, let's let OK, our graph Y be equal to F of X. Then if our graph is symmetric about the X axis, OK, then FFX is going to be equal to negative F of X. If it's symmetric about the Y axis, then FFX would be equal to F of negative X. And if it's symmetric about the origin, then F of X would be equal to the negative value of FF negative X. In other words, when we change our values of X and Y in our original equation that can help us to figure out where it would be symmetric. So let's test all of these three conditions to see which one it fits. Let's start by testing if it's symmetric about the X axis. Now here, if we change OK. Our equation here, Y if we replace Y with negative Y, OK, then negative F of X or negative Y is going to be equal to the negative value of X to the sixth plus seven X squared minus nine. When we simplify that negative X to the sixth minus seven X squared plus nine is that value equal to our original function. No, it's not equal to Y or F FX. Therefore, this tells us that our graph, OK. Let me just clean that up here. Or graph is not symmetric about the X axis. OK. So we can take that one off of our list. Let's cross it out next. What about Y axis? Well, no, for the Y axis, we need to figure out if FF minus X equals F of X. So in other words, we're going to substitute negative X for X through our equation. So that's going to be negative X to the sixth plus seven multiplied by negative X squared minus nine, negative X to the sixth equals X to the sixth, negative X squared equals X squared. So that's gonna be plus seven X squared minus nine. Notice here that F of negative X is indeed equal to F of X. This is true. Therefore, that tells us then that or graph, OK. Or graph is symmetric about the wire access So we know that that condition is true. Let's test the third one is our graph symmetric about the origin. Well, that means we're gonna find negative F of negative X. OK. So this is gonna be equal to the negative value of negative X to the sixth plus seven multiplied by negative X squared OK minus nine. And now again, when we work out what's inside our bracket here, we already know that F of negative X equals F FX. So that's negative X to the sixth plus seven X squared minus nine. And then we know that this is gonna be equal to negative X to the sixth minus seven X squared plus nine. And that value is not equal to F of X. Therefore, that tells us our graph, our graph is not symmetric about the origin. So now we know that our graph is symmetric about the Y axis. If we look back at our answer choices that tells us that B is the correct answer. Thanks a lot 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.
$ƒ (x) = x^{4} + 5 x^{2} - 12$

Key Concepts

Symmetry in Graphs

Even and Odd Functions

Graphing Functions

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