I need help solving the two problems below. Thanks
For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin.
Check all symmetries that apply.
1. y=-sqrt(4-x^(2))
2. 34x^(2)+12y^(2)=18
1.With each point (x,y) we also have point
(-x,y)-->the graph is symmetric y-axis
2.With each point (x,y) we have points
(-x,y), (x,-y), (-x,-y) -->
y-axis, x-axis, origin
Thanks!!!!
Sure! Let's start by solving the first problem.
1. y = -sqrt(4 - x^2)
To determine the symmetry of the graph with respect to the x-axis, you can replace y with -y in the equation and see if it remains the same. Let's substitute -y for y:
-y = -sqrt(4 - x^2)
To simplify, let's multiply both sides of the equation by -1:
y = sqrt(4 - x^2)
Notice that we obtained the original equation, so the graph is symmetric with respect to the x-axis.
To determine the symmetry with respect to the y-axis, you can replace x with -x in the equation and see if it remains the same. Let's substitute -x for x:
y = -sqrt(4 - (-x)^2)
Simplifying the equation:
y = -sqrt(4 - x^2)
This is the same as the original equation, so the graph is symmetric with respect to the y-axis as well.
To determine symmetry with respect to the origin, we need to check if replacing both x with -x and y with -y results in the same equation. Let’s substitute -x and -y into the equation:
-y = -sqrt(4 - (-x)^2)
Simplifying:
-y = -sqrt(4 - x^2)
Rearranging the equation:
y = sqrt(4 - x^2)
This equation is different from the original one, so the graph is not symmetric with respect to the origin.
Now let's move on to the second problem.
2. 34x^2 + 12y^2 = 18
Again, let's test for symmetry with respect to the x-axis by replacing y with -y in the equation:
34x^2 + 12(-y)^2 = 18
Simplifying:
34x^2 + 144y^2 = 18
This equation is not the same as the original, so the graph is not symmetric with respect to the x-axis.
For symmetry with respect to the y-axis, we replace x with -x:
34(-x)^2 + 12y^2 = 18
Simplifying:
34x^2 + 12y^2 = 18
This equation is identical to the original, so the graph is symmetric with respect to the y-axis.
To test for symmetry with respect to the origin, we need to replace x with -x and y with -y:
34(-x)^2 + 12(-y)^2 = 18
Simplifying:
34x^2 + 12y^2 = 18
Again, this equation is identical to the original, so the graph is also symmetric with respect to the origin.
Hope that helps! Let me know if you have any further questions.