Q. 47 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry -

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

Step-by-Step Solution

Answer

• $(0,0)$ is both the x and y intercept.
• The graph is not symmetric about x-axis.
• The graph is not symmetric about y-axis.
• The graph is symmetric about the origin.

Step 1. Given Data

Equation -  $y=\sqrt[3]{x}$

Step 2. To Find

Find the intercept point and test for symmetry.

Step 3. Finding the x-intercept

To find the x-intercepts, replace y with 0 and solve for x.

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

Put  $y=0$

$0=\sqrt[3]{x}$

$x=0$

(0,0) is the x-intercept. 

Step 4. Finding the y-intercept

To find the x-intercepts, replace x with 0 and solve for y.

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

Put  $x=0$

$y=\sqrt[3]{0}$

$y=0$

(0,0) is the x-intercept. 

Step 5. Check for symmetry

• Checking symmetry about x-axis

if replacing (y) with (-y) leaves the equation unchanged, the graph will be symmetric about the x-axis.

we will start with original equation -

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

Replacing (y) with (-y) gives us -  

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

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

This is not the same as the original equation, the graph will not be symmetric about the x-Axis.

• Checking symmetry about y-axis

if replacing (x) with (-x) leaves the equation unchanged, the graph will be symmetric about the y-axis.

we will start with original equation -

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

Replacing (x) with (-x) gives us -  

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

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

This is not the same as the original equation, the graph will not be symmetric about the y-Axis.

• Checking symmetry about the origin

if replacing (x) with (-x) and (y) with (-y)leaves the equation unchanged, the graph will be symmetric about the origin.

we will start with original equation -

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

Replacing (x) with (-x) and (y) with (-y) gives us -  

$(-y)=\sqrt[3]{(-3)}$

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

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

This is same as the original equation, the graph will be symmetric about the origin.

Step 6. Graph

The graph of equation $y=\sqrt[3]{x}$ will be -