Answer
1) The $x$-intercept is $(3,0)$, and the $y$-intercept is $(0,-27)$.
2) The graph of the equation $y={{x}^{3}}-27$ is not symmetric with respect to the $x$-axis, $y$-axis, and the origin.
Work Step by Step
To find $x$-intercept(s), let$ y=0$
$\Rightarrow \,\,0={{x}^{3}}-27$
$\Rightarrow \,\,{{x}^{3}}=\,27$
$\Rightarrow \,\,x\,=\,3$
Therefore, the $x$-intercept is $(3,0)$
To find $y$-intercept(s), let $ x=0$
$\Rightarrow \,\,y=0-27$
$\Rightarrow \,\,y=-27$
The $y$-intercept is $(0,-27)$
To test for symmetry with respect to the $x$-axis, replace $y$ with $-y$. If the result is equivalent to
the original equation, then the equation is symmetric with respect to the $x$-axis.
$\Rightarrow (-y)={{x}^{3}}-27$ is not equivalent to the equation $y={{x}^{3}}-27$.
Hence, the graph of the equation $y={{x}^{3}}-27$ is not symmetric with respect to the $x$-axis
To test for symmetry with respect to the $y$-axis, replace $x$ with$-x$. If the result is equivalent to
the original equation, then the equation is symmetric with respect to the $y$-axis.
$\Rightarrow y={{(-x)}^{3}}-27$ is not equivalent to the equation $y={{x}^{3}}-27$.
Hence, the graph of the equation $y={{x}^{3}}-27$ is not symmetric with respect to the $y$-axis.
To test for symmetry with respect to the origin, replace $x$ wutg $x$ and $y$ with $-y$. If the result is equivalent to the original equation, then the equation is symmetric with respect to the origin.
$\Rightarrow \,-y={{(-x)}^{3}}-27$ is not equivalent to the equation $y={{x}^{3}}-27$
Hence, the graph of the equation $y={{x}^{3}}-27$ is not symmetric with respect to the origin.
