Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 53, list the intercepts and test for symmetry.

$y=x^3-27$

Step-by-Step Solution

Answer

The graph of the equation is not symmetric with respect to the $x$-axis, $y$-axis and the origin, and the $x$-intercept is $(3,0)$

The $y$-intercept is $(0,-27)$.

Step 1: Given information

The given function is $y=x^3-27$. 

Step 2: Determine the x -intercept.

For $x$-intercept, substitute $0$ for $y$ and then solve for $x$.

$\begin{array}{rcl}0& =& x^3-27\\ x^3& =& 27\\ x^3& =& 3^3\\ x& =& 3\end{array}$

Step 4: Determine the y -intercept.

For $y$-intercept, substitute 0 for $x$ and then solve for $y$.

$\begin{array}{rcl}y& =& 0^3-27\\ y& =& -27\end{array}$

Step 4: Determine the symmetry.

Substitute $y$ for $-y$ in the equation to check for symmetry with respect to $x$ axis:

$$\begin{gathered} -y=x^3-27 \\ y=-x^3+27 \end{gathered}$$

Notice that the right side of the equation after substitution is not equal to the right side of the initial equation.

Since the equations are not equivalent, the graph is not symmetric with respect to the $x$-axis.

Substitute $x$ with $-x$ in the equation to check for the symmetry with respect to $y$-axis:

$\begin{array}{rcl}y& =& {\left(-x\right)}^3-27\\ & =& -x^3-27\end{array}$

Notice that the right side of the equation after substitution is not equal to the right side of the initial equation.

Since the equations are not equivalent, the equation is not symmetric with respect to the $y$-axis.

Substitute $x$ with $-x$ and $y$ with $-y$ in the equation to check for the symmetry with respect to the origin:

$\begin{array}{rcl}-y& =& {\left(-x\right)}^3-27\\ -y& =& -x^3-27\\ y& =& x^3+27\end{array}$

Notice that the right side of the equation after substitution is not equal to the right side of the initial equation.

Since the equations are not equivalent, the equation is not symmetric with respect to the origin.

Step 5: Write the conclusion.

Therefore, the $x$-intercept is $\left(3,0\right)$ and $y$-intercept is $\left(0,-27\right)$.

The given equation is not symmetric with respect to $x$-axis, $y$-axis and the origin.