Question

use the algebraic tests to check for symmetry with respect to both axes and the origin. (select all that apply.)
$x - y^2 = 19$
$square$ x - axis symmetry
$square$ y - axis symmetry
$square$ origin symmetry
$square$ no symmetry

Explanation:

Step1: Test x - axis symmetry

To test for x - axis symmetry, we replace \(y\) with \(-y\) in the equation \(x - y^{2}=19\).
Substituting \(y=-y\) into the equation, we get \(x-(-y)^{2}=x - y^{2}\) (since \((-y)^{2}=y^{2}\)). So the equation becomes \(x - y^{2}=19\), which is the same as the original equation. So the graph is symmetric about the x - axis.

Step2: Test y - axis symmetry

To test for y - axis symmetry, we replace \(x\) with \(-x\) in the equation \(x - y^{2}=19\).
Substituting \(x = -x\) into the equation, we get \(-x-y^{2}=19\), which is not the same as the original equation \(x - y^{2}=19\). So the graph is not symmetric about the y - axis.

Step3: Test origin symmetry

To test for origin symmetry, we replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(x - y^{2}=19\).
Substituting \(x=-x\) and \(y = -y\) into the equation, we get \(-x-(-y)^{2}=-x - y^{2}\). This is not the same as the original equation \(x - y^{2}=19\). So the graph is not symmetric about the origin.

Answer:

x - axis symmetry