Restated:For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin.
4y^4+24x^2=28 and y=1/(x^2+5)
Check all symmetries that apply
x-axis,y-axis,origin or non of the above.
thanks
Please check your 1st post.
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To determine the symmetry of a graph with respect to the x-axis, y-axis, or the origin, we need to analyze the equation and substitute negative values for either x or y to see if the equation remains unchanged.
1. For the equation 4y^4 + 24x^2 = 28:
- To check for symmetry with respect to the x-axis, replace y with -y:
4(-y)^4 + 24x^2 = 28
Simplifying further: 4y^4 + 24x^2 = 28
Since both equations are the same, the graph is symmetric with respect to the x-axis.
- To check for symmetry with respect to the y-axis, replace x with -x:
4y^4 + 24(-x)^2 = 28
Simplifying further: 4y^4 + 24x^2 = 28
Similar to the previous case, the equation remains the same, indicating symmetry with respect to the y-axis.
- To check for symmetry with respect to the origin, replace both x and y with their negative counterparts:
4(-y)^4 + 24(-x)^2 = 28
Simplifying further: 4y^4 + 24x^2 = 28
As the equation is unchanged again, the graph is symmetric with respect to the origin.
2. For the equation y = 1/(x^2 + 5):
- To check for symmetry with respect to the x-axis, replace y with -y:
-y = 1/(x^2 + 5)
Multiply both sides by -1 to maintain equality:
y = -1/(x^2 + 5)
Since the equation is different, the graph is not symmetric with respect to the x-axis.
- To check for symmetry with respect to the y-axis, replace x with -x:
y = 1/((-x)^2 + 5)
Simplifying further: y = 1/(x^2 + 5)
Once again, the equation remains the same, indicating symmetry with respect to the y-axis.
- To check for symmetry with respect to the origin, replace both x and y with their negative counterparts:
-y = 1/((-x)^2 + 5)
Simplifying further: -y = 1/(x^2 + 5)
Since the equation is different, the graph is not symmetric with respect to the origin.
In summary:
- The equation 4y^4 + 24x^2 = 28 is symmetric with respect to the x-axis, the y-axis, and the origin.
- The equation y = 1/(x^2 + 5) is only symmetric with respect to the y-axis. It is not symmetric with respect to the x-axis or the origin.