You can test for x or y symmetry, respectively, by replacing the variable name opposite of what symmetry you're looking for with something negative. If the equation is the same after plugging in, then it's symmetric.
\(ex.\,\,a.\)
\(Does\,\,this\,\,parabola\,\,have\,\,x\,\,symmetry?\)
\(y = 5x^{2} - 2 \)
\(-y = 5x^{2} - 2\)
\(\dfrac{-y}{-1} = \dfrac{5x^{2}}{-1} - \dfrac{2}{-1}\)
\(y = -5x^{2} + 2\)
\(No. \,\,Above\,\,equation\,\,is\,\, a \,\,different \,\,equation.\)
\(ex.\,\,b\)
\(Does\,\,this\,\,parabola\,\,have\,\,y\,\,symmetry?\)
\(y = -x^{2} + 14\)
\(y = -(-x)^{2} + 14\)
\(y = -x^{2} + 14 \)
\(Yes.\,\,The\,\,equation\,\,is\,\,the\,\,same\,\,equation.\)
Replacing both x and y with negative equivalents let's us test for origin symmetry because if the equation doesn't change, it has it.
\(ex.\,\,c.\)
\(Does\,\,this\,\,parabola\,\,have\,\,origin\,\,symmetry?\)
\(y = 13x^{2} - 6x + 1\)
\(-y = 13(-x)^{2} - 6(-x) + 1\)
\(-y = 13x^{2} + 6x + 1\)
\(\dfrac{-y}{-1} = \dfrac{13x^{2}}{-1} + \dfrac{6x}{-1} + \dfrac{1}{-1}\)
\(y = -13x^{2} - 6x - 1\)
\(No.\,\,The\,\, above\,\,equation \,\,is\,\, different.\)
