Part a: Writing the Function

To solve for the function $g(x)$, we need to account for the horizontal and vertical translations of the cube root function.

The general form of a cube root function is:

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

For transformations, the formula is:

$g(x) = a \cdot \sqrt[3]{b(x - h)} + k$

Where:

• $a$ controls vertical stretch or compression (doesn't apply in this case).
• $b$ controls horizontal stretch or compression.
• $h$ is the horizontal translation.
• $k$ is the vertical translation.

Given:

• The vertical translation $k$ is negative.
• The horizontal translation $h$ is three times the vertical translation $k$. Therefore, $h = 3k$.

Thus, we have the relationships: $h = 3k \quad \text{and} \quad k < 0$

Let $k = -1$, then $h = 3(-1) = -3$.

The function would then be:

[ g(x) = \sqrt[3]{x - (-3)} - 1 = \sqrt[3]{x + 3} - 1 ]

Thus, the function is:

$g(x) = \sqrt[3]{x + 3} - 1$

Part b: Finding the Correct Option

Now, let's check the given options to see which one matches this equation:

• Option A: $g(x) = \sqrt[3]{x - 5} + 2$

  • This does not match because the horizontal translation should be $+3$, not $-5$, and the vertical translation should be $-1$, not $+2$.

• Option B: $g(x) = \sqrt[3]{x + 3} - 1$

  • This is exactly what we derived, so it’s the correct option.

• Option C: $g(x) = \sqrt[3]{x - 3} + 1$

  • This does not match because the horizontal translation is $-3$, not $+3$, and the vertical translation is $+1$, not $-1$.

• Option D: $g(x) = \sqrt[3]{x + 5} - 2$

  • This does not match because the horizontal translation is $+5$, not $+3$, and the vertical translation is $-2$, not $-1$.

Therefore, the correct answer is Option B.

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Questions to Expand:

1. How would the graph of the parent function $f(x) = \sqrt[3]{x}$ look before and after applying horizontal and vertical translations?
2. What would happen if the vertical translation was positive instead of negative? How would this affect the graph?
3. How can you determine the direction of the shift (left or right, up or down) based on the transformations?
4. How does the horizontal translation affect the domain and range of a cube root function?
5. What would the graph look like if the horizontal translation were three times the opposite of the vertical translation (i.e., $h = -3k$)?

Tip:

Remember that for cube root functions, both horizontal and vertical shifts maintain the basic shape of the curve but simply translate the graph in the respective directions.