Chapter 1.2, Problem 57AYU

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

$y=x^3-27$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 57AYU

$x\text{-intercept}$ is 3 and $y\text{-intercepts}$ is $-27$. 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 $y = x^3 - 27$

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

Therefore   $0 = x^3 - 27$

$0+27 = x^3-27+27$   Add 27 on both sides

$x^3 = 27$

$x^{} = 3$

The $x\text{-intercepts}$ is 3

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

Therefore   $y = {(0)}^3 - 27$

$y=-27$

The $y\text{-intercepts}$ is $-27$

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

$\text{(<mo>−</mo><mi>y</mi>)} = x^3 - 27$

$-y = x^3 - 27$ is 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 $y\text{-axis}$. Replace $x$ by $-x$.

$y = {(-x)}^3 - 27$

$y = -x^3 - 27$ is not equivalent to the original equation.

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

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

$\text{(<mo>−</mo><mi>y</mi>)} = {(-x)}^3 - 27$

$-y = -x^3 - 27$ is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.