Q. 54 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 54, list the intercepts and test for symmetry.

$y=x^4-1$

Step-by-Step Solution

Answer

The equation is symmetric with respect to the $y$-axis, and the points of the equation $(-1,0),(1,0)$ are the $x$-intercepts and $(0,-1)$ is the $y$-intercept.

Step 1: Given information:

The given equation is $y=x^4-1$.

Step 2: Determine the x -intercept.

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

$\begin{array}{rcl}0& =& x^4-1\\ \left(x^2-1\right)\left(x^2+1\right)& =& 0\\ x^2-1& =& 0\\ \left(x-1\right)\left(x+1\right)& =& 0\\ x& =& 1,-1\end{array}$

Step 3: Determine the y -intercept.

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

$\begin{array}{rcl}y& =& 0^4-1\\ & =& -1\end{array}$

Step 4: Determine the symmetry

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

$\begin{array}{rcl}-y& =& x^4-1\\ y& =& -x^4+1\end{array}$

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 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:

$\begin{array}{rcl}y& =& {\left(-x\right)}^4-1\\ & =& x^4-1\end{array}$

Notice that the right side of the equation after substitution is equal to the right side of the initial equation.

Since the equations are equivalent, the equation is symmetric with respect to the $y$-axis.

Step 5: Write the conclusion

Therefore, the $x$-intercept is $\left(1,0\right)$ and $\left(-1,0\right)$.

The $y$-intercept is $\left(0,-1\right)$.

The given equation is symmetric with respect to y-axis.