Tell if the function is symmetric with respect to the x-axis, y-axis, and origin.
3. 8x^2 - 2y^2 = 12
==> I got that it was symmetric to both the x- and y-axes. Is this right?
4. 5x^2 - 3y^2 = 18
==> I can't get anywhere with this equation -- I'm getting the answer that this equation is imaginary.
Can someone please show me how to do these? Thanks soooo much :D
To determine if a function is symmetric with respect to the x-axis, y-axis, or origin, we need to evaluate the function when we replace x with -x, y with -y, or both x and y with their negative values and see if the equation still holds.
Let's start with the first equation:
3(8x^2 - 2y^2) = 12
To check for symmetry with respect to the x-axis, replace y with -y:
3(8x^2 - 2(-y)^2) = 12
3(8x^2 - 2y^2) = 12
Since the equation remains the same when we replace y with -y, it is symmetric with respect to the x-axis.
To check for symmetry with respect to the y-axis, replace x with -x:
3(8(-x)^2 - 2y^2) = 12
3(8x^2 - 2y^2) = 12
Again, the equation remains the same, so it is symmetric with respect to the y-axis as well.
Next, to check for symmetry with respect to the origin, replace both x and y with their negative values:
3(8(-x)^2 - 2(-y)^2) = 12
3(8x^2 - 2y^2) = 12
Once again, the equation remains the same, so it is symmetric with respect to the origin.
Therefore, the first equation is symmetric with respect to the x-axis, y-axis, and origin.
Moving on to the second equation:
5x^2 - 3y^2 = 18
To check for symmetry with respect to the x-axis, replace y with -y:
5x^2 - 3(-y)^2 = 18
5x^2 - 3y^2 = 18
The equation remains the same, so it is symmetric with respect to the x-axis.
To check for symmetry with respect to the y-axis, replace x with -x:
5(-x)^2 - 3y^2 = 18
5x^2 - 3y^2 = 18
Once again, the equation remains the same, so it is symmetric with respect to the y-axis.
However, to check for symmetry with respect to the origin, we need to replace both x and y with their negative values:
5(-x)^2 - 3(-y)^2 = 18
5x^2 - 3y^2 = 18
As we can see, the equation remains the same, so it is symmetric with respect to the origin as well.
Therefore, the second equation is symmetric with respect to the x-axis, y-axis, and origin, contrary to your intuition that it is imaginary.
It's important to note that finding the symmetry of an equation is a step-by-step process. Sometimes the equation might simplify or transform in a way that allows us to see the symmetry more easily. In this case, the equations already had the symmetry form, making it easier to determine the symmetries.