How would you determine whether |y|= x^3 is symmetric with respect to the x-axis, y-axis, both, or neither.
In my book it said to plug in (a,b) and to get an equation to compare the other ones to. So I got |b|= a^3. Then I plugged in (a,-b) to test the x-axis and (-a,b) to test the y-axis. So I got |-b|= a^3 and |b|= -a^3. So then I though that it would be symmetrical with respect to the x-axis, but when I graphed it, it was symmetrical with respect to the y-axis.
One way to do it is to draw the graph. That would be rather instructive.
Clearly it not symmetric about the y axis because x cannot be negative. The lest side of the equation is always positive or zer.
For x > 0, there are two possible y values for each x (+x^3 and -x^3) , and they are symmetric about the x axis.
Why x can't be negative? Isn't it just y that can't be negative?
If x is negative , you'd have a negative number on the right and a positive number (an absolute value) on the left. That cannot happen.
Try graphing it and you will see.
To determine the symmetry of the equation |y| = x^3, it is essential to consider how the equation behaves when certain substitutions are made.
Let's start by systematically testing for symmetry with respect to the x-axis and the y-axis.
1. Testing for symmetry with respect to the x-axis:
First, substitute (a,b) into the equation |y| = x^3, which gives |b| = a^3. Next, substitute (a,-b) into the given equation, which results in |-b| = a^3. Simplifying the equation |-b| = a^3, we get |b| = a^3, which is the same as the equation obtained when substituting (a,b). This indicates that the equation is symmetric with respect to the x-axis.
2. Testing for symmetry with respect to the y-axis:
Using the same approach, substitute (a,b) into the equation |y| = x^3, giving |b| = a^3. Then, substitute (-a,b) into the equation, yielding |b| = (-a)^3, which simplifies to |b| = -a^3. Comparing |b| = a^3 and |b| = -a^3, we can observe that they are not equal. Therefore, the equation is not symmetric with respect to the y-axis.
Based on the results of the tests, we can conclude that the equation |y| = x^3 is symmetric only with respect to the x-axis and not with respect to the y-axis.
In your case, you mentioned that when you graphed the equation, it appeared to be symmetrical with respect to the y-axis. It is possible that there might have been an error in graphing or a misunderstanding of the symmetry concept. Double-checking the graphing process may be helpful in resolving any discrepancies.