Question

without graphing, determine whether the following equation has a graph that is symmetric with respect to the x - axis, the y - axis, the origin, or none of these.
$x^2 + y^2 = 12$
select all that apply
a. origin
b. x - axis
c. y - axis
d. none of these

Explanation:

Step1: Test x-axis symmetry

Replace $y$ with $-y$:
$x^2 + (-y)^2 = 12 \implies x^2 + y^2 = 12$
This matches the original equation, so symmetric to x-axis.

Step2: Test y-axis symmetry

Replace $x$ with $-x$:
$(-x)^2 + y^2 = 12 \implies x^2 + y^2 = 12$
This matches the original equation, so symmetric to y-axis.

Step3: Test origin symmetry

Replace $x$ with $-x$ and $y$ with $-y$:
$(-x)^2 + (-y)^2 = 12 \implies x^2 + y^2 = 12$
This matches the original equation, so symmetric to origin.

Answer:

A. origin
B. x-axis
C. y-axis