Chapter 1.2, Problem 64AYU

In Problems 49-64, list the intercepts and test for symmetry.

$y=x^4-1$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 64AYU

$x\text{-intercept}$ are $-1$ and 1 and $y\text{-intercept}$ is $-1$. The graph of the equation is symmetric with respect to $y\text{-axis}$.

Explanation of Solution

Given:

The equation $y = x^4-1$

To find the $x\text{-intercept(s)}$, let $y=0$

Therefore             $0=x^4-1$

$x^4-1=0$

$(x^2-1)(x^2+1)=0$

$x^2-1=0$ or $x^2+1=0$

$x^2=1$ or $x^2=-1$

$x=\pm 1$ or $x=\pm \sqrt{-1}$ not real

The $x\text{-intercept}$ are $-1$ and 1

To find the $y\text{-intercept(s)}$, let $x=0$

Therefore   $y={(0)}^4-1$

$y=-1$

The $y\text{-intercept}$ is $-1$

Now test the equation for symmetry with respect to $x\text{-axis}$. Replace $y$ by $-y$.

$(-y)=x^4-1$

$-y=x^4-1$not the same as the original equation when $y$ is replaced by $-y$.

Therefore the graph of the equation is not symmetric with respect to $x\text{-axis}$.

Now test the equation for symmetry with respect to $-y$. Replace $x$ by $-x$

$y={(-x)}^4-1$

$y=x^4-1$is not equivalent to the original equation.

Therefore the graph of the equation is not symmetric with respect to $y\text{-axis}$.

Now test the equation for symmetry with respect to the origin. Replace $x$ by $-x$ and $y$ by $-y$

$(-y)={(-x)}^4-1$

$-y=x^4-1$ is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.