Question

List the intercepts and test for symmetry. $$ y=\sqrt[3]{x} $$

Short answer

Intercept: (0,0). It is symmetric with respect to the origin only.

Step-by-step solution

- Identify the Intercepts

To identify the intercepts, find where the graph of the function crosses the x-axis and y-axis. For the x-intercept, set \(y = 0\) and solve for \(x\).\[0 = \sqrt[3]{x} \Rightarrow x = 0.\]For the y-intercept, set \(x = 0\) and solve for \(y\).\(y = \sqrt[3]{0} = 0\). So, the function only has one intercept at (0,0).

- Test for Symmetry with Respect to the Y-Axis

To test if the graph is symmetric with respect to the Y-axis, replace \(x\) with \(-x\) and compare it to the original equation: \[y = \sqrt[3]{-x}.\]Since \(\sqrt[3]{-x} eq \sqrt[3]{x}\), the function is not symmetric with respect to the y-axis.

- Test for Symmetry with Respect to the X-Axis

To test if the graph is symmetric with respect to the X-axis, replace \(y\) with \(-y\) and solve for \(x\): \[-y = \sqrt[3]{x}\]. Cubing both sides yields \[-y^3 = x\], which is not equivalent to the original equation. Therefore, the function is not symmetric with respect to the x-axis.

- Test for Symmetry with Respect to the Origin

To test if the graph is symmetric with respect to the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\): \[-y = \sqrt[3]{-x}\]. Cubing both sides gives \[-y^3 = -x\] or \(y^3 = x\), which is equivalent to \(y = \sqrt[3]{x}\). Therefore, the function is symmetric with respect to the origin.

Key concepts

x-intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. To find it, set y to 0 and solve for x. In our example with the function \( y = \sqrt[3]{x} \), we set y to 0 and solve for x as follows: \[ 0 = \sqrt[3]{x} \Rightarrow x = 0. \] This tells us that the x-intercept is at the point (0,0). If you graph different equations, you'll notice that intercepts can occur at different points, but the process remains the same: set y to 0 and solve for x.

y-intercept

The y-intercept is where the graph of a function crosses the y-axis. To find it, set x to 0 and solve for y. For the function \( y = \sqrt[3]{x} \), we do this calculation: \[ y = \sqrt[3]{0} = 0. \] So, the y-intercept is also at the point (0,0). Similar to finding the x-intercept, setting x to 0 and solving for y in other equations will help identify where other functions cross the y-axis.

graph symmetry

Graph symmetry can help us understand the shape and behavior of a function. There are three types to consider: y-axis, x-axis, and origin symmetry.

• To test for y-axis symmetry, replace x with -x and see if the equation remains unchanged. For our function, since \sqrt[3]{-x} \e \sqrt[3]{x}, it is not symmetric about the y-axis.
• Checking for x-axis symmetry involves replacing y with -y and solving for x. For \( y = \sqrt[3]{x} \), transforming it to \ -y = \sqrt[3]{x} \ and cubing both sides gives \ -y^3 \e x , hence no x-axis symmetry.
• Lastly, for origin symmetry, replace x with -x and y with -y. If both changes lead back to the initial equation, the function is symmetric about the origin. Our example transforms to \ -y = \sqrt[3]{-x} \ and then cubing both sides gives \ y^3 = x, which matches our original equation. Hence, \( y = \sqrt[3]{x} \) is symmetric about the origin.

Understanding symmetry can provide valuable insights into how a function behaves and help simplify graphing and analysis.