Precalculus, Chapter 1, 1.2, Section 1.2, Problem 64

Find the intercepts of the equation $y = x^4 - 1$ and test for symmetry.

$x$-intercepts:


$
\begin{equation}
\begin{aligned}

y =& x^4 - 1
&& \text{Given equation}
\\
0 =& x^4 - 1
&& \text{To find the $x$-intercept, we let } y = 0
\\
1 =& x^4
&&
\\
\pm 1 =& x
&&

\end{aligned}
\end{equation}
$



The $x$-intercepts are $(-1,0)$ and $(1,0)$

$y$-intercepts:


$
\begin{equation}
\begin{aligned}

y =& x^4 - 1
&& \text{Given equation}
\\
y =& (0)^4 - 1
&& \text{To find the $y$-intercept, we let } x = 0
\\
y =& -1
&&

\end{aligned}
\end{equation}
$


The $y$-intercept is $(0,-1)$

Test for symmetry

$x$-axis:


$
\begin{equation}
\begin{aligned}

y =& x^4 - 1
&& \text{Given equation}
\\
-y =& x^4 - 1
&& \text{To test for $x$-axis symmetry, replace $y$ by $-y$ and see if the equation is still the same}

\end{aligned}
\end{equation}
$


The equation changes so it is not symmetric to the $x$-axis

$y$-axis:


$
\begin{equation}
\begin{aligned}

y =& x^4 - 1
&& \text{Given equation}
\\
y =& (-x)^4 - 1
&& \text{To test for $y$-axis symmetry, replace$ x$ by $-x$ and see if the equation is still the same}
\\
y =& x^4 - 1
&&

\end{aligned}
\end{equation}
$


The equation is still the same so it is symmetric to the $y$-axis

Origin:


$
\begin{equation}
\begin{aligned}

y =& x^4 - 1
&& \text{Given equation}
\\
-y =& (-x)^4 - 1
&& \text{To test for origin symmetry, replace both $x$ by $-x$ and y by $-y$ and see if the equation is still the same}
\\
-y =& x^4 - 1
&&

\end{aligned}
\end{equation}
$


The equation changes so it is not symmetric to the origin.

Therefore, the equation $y = x^4 - 1$ has an intercepts $(-1,0), (1,0)$ and $(0,-1)$ and it is symmetric to the $y$-axis.

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