In Exercises 5–8, determine whether the graph of the function ...

Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x² - 2x - 1

Accepted Answer

Welcome back everyone. For the given function below is a symmetric about the y axis, the origin both or neither. Y equals X2 + 3 X minus 5. For this problem, let's follow simple steps. Let's begin by defining the original function F of X. From there our next step is to evaluate f of negative X, and then we're going to expand two conditions. If F of negative X is actually equal to the original function F of X, we can conclude that our function is even, and if it is even, then it's symmetric about the y axis. On the other hand, if f of negative X becomes equal to negative of F of X, then we are dealing with an odd function. And an odd function is symmetric relative to the origin. If none of these conditions are met, then we can simply say that it's neither. So for this problem, we're just going to follow those steps. Our F of X is equal to X2 + 3X minus 5. We want to evaluate the F of negative X. Which is negative X2. Plus 3 multiplied by negative X minus 5. If we square a negative number, it becomes positive, so we get X2 minus 3X minus 5. And now we're going to compare it to the original function to begin with, which is X2 + 3x minus 5. We can see that those are not equal to each other due to the negative sign. In front of 3 on the left and the positive sign in front of 3 on the right. So because they're not equal to each other, it is not even, meaning not symmetric relative to the y axis. Now what we're going to do from here is essentially factor out the negative sign. So F of negative X. is equal to negative and now incs we can simply add minus x2 + 3 x + 5. Simply speaking, we're just flipping all of those signs, right, because we're factoring out a negative sign. And now what we want to do to make sure that this function is odd. Is that the parties should be F of X by definition, right? Negative F of X, right? Because if we are evaluating F of negative X. It must be equal to negative f of x. Now, is this part negative of F of X? Well, it is not because by simple observation we can see that we have a negative sign in front of X2d while in the original function we have positive X2, and additionally we have a positive sign in front of 5 instead of the original negative sign. So we can see that. F of negative X is not equal to negative of F of X, meaning it is not odd. And if it's not even, not odd, it means it's neither. So it's neither symmetric about the y-axis nor the origin. We can just say neither, which is our final answer to this problem. Thank you for watching.

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x² - 2x - 1

Key Concepts

Symmetry about the y-axis

Symmetry about the origin

Analyzing polynomial functions

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