Chapter 1.2, Problem 51AYU

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

$y=\sqrt[3]{x}$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 51AYU

$x\text{-intercept}$ is 0 and $y\text{-intercepts}$ are 0. The graph of the equation is not symmetric with respect to $x\text{-axis}$, $y\text{-axis}$ and the origin.

Explanation of Solution

Given:

The equation $y\text{ = }\sqrt[3]{x}$

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

Therefore   $\sqrt[3]{x}=0$

$x\text{ = 0}$

The $x\text{-intercept}$ is 0

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

Therefore   $y\text{ =}\sqrt[3]{0}$

$y\text{ = 0}$

The $x\text{-intercept}$ is 0 and $y\text{-intercept}$ is 0

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

$\left(-y\right)\text{ =}\sqrt[3]{x}$

$\text{<mo>−</mo><mi>y</mi> = }\sqrt[3]{x}$. The equation is not same 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 $y\text{-axis}$. Replace $x$ by $-x$.

$y\text{ = }\sqrt[3]{-}x$ is not equivalent to $y\text{ = }\sqrt[3]{x}$.

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

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

$\text{<mo>−</mo><mi>y</mi> = }\sqrt[3]{-}x$ which is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.