Question

question 19 points 2

assume that the values of \\(h\\) and \\(k\\) are greater than 0. if \\(y = a\sqrt3{x - h} + k\\) has the value of \\(a = 1\\), the graph will ______.

• be closer to the \\(y\\)-axis
• translate the parent function \\(h\\) units right and \\(k\\) units up

Explanation:

Identify the parent function and parameters

The given equation is \(y = a\sqrt[3]{x - h} + k\).
The parent function is the Cube Root Function, \(f(x) = \sqrt[3]{x}\).
We are given that \(a = 1\), and both \(h > 0\) and \(k > 0\).
Substituting \(a = 1\) into the equation yields:

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

Apply horizontal translation

Using the principles of Function Transformations, replacing \(x\) with \(x - h\) shifts the graph horizontally.
Since \(h > 0\), subtracting \(h\) inside the function shifts the graph to the right by \(h\) units.

Apply vertical translation

Adding \(k\) to the outside of the function shifts the graph vertically.
Since \(k > 0\), adding \(k\) shifts the graph upward by \(k\) units.

Combine the transformations

Combining these two shifts, the graph of the parent function \(y = \sqrt[3]{x}\) is translated \(h\) units to the right and \(k\) units up.
This matches the second option: "translate the parent function \(h\) units right and \(k\) units up".

Answer:

• be closer to the \(y\)-axis
• translate the parent function \(h\) units right and \(k\) units up (Correct answer)