Question

graphing the cube root function practice
complete this assessment to review what youve learned. it will not count toward your grade.
select the statement that best describes the end behavior of the function ( h(x) = sqrt3{x - 4} + \frac{1}{3} )
statement #1: as x approaches infinity, y approaches negative infinity.
statement #2: as x approaches negative infinity, y approaches infinity.
statement #3: as x approaches infinity, y approaches infinity.
(1 point)
statement #□ best describes the end behavior of the function.

Explanation:

Step1: Recall cube root function end behavior

The parent cube root function \( y = \sqrt[3]{x} \) has the end behavior: as \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to -\infty \).

Step2: Analyze the transformed function \( h(x)=\sqrt[3]{x - 4}+\frac{1}{3} \)

The transformation \( x\to x - 4 \) is a horizontal shift (does not affect end behavior), and \( +\frac{1}{3} \) is a vertical shift (also does not affect end behavior). So the end behavior of \( h(x) \) is the same as \( y=\sqrt[3]{x} \).

• For \( x\to\infty \): \( \sqrt[3]{x - 4}\to\infty \), so \( h(x)=\sqrt[3]{x - 4}+\frac{1}{3}\to\infty \) (matches Statement #3).
• For \( x\to-\infty \): \( \sqrt[3]{x - 4}\to-\infty \), so \( h(x)\to-\infty \) (so Statement #2 is wrong).
• Statement #1 says as \( x\to\infty \), \( y\to-\infty \), which is wrong.

Answer:

Statement #3: As x approaches infinity, y approaches infinity.