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The given function is:

\[
y = 2 \sqrt[3]{x - 8} + 2
\]

To find the **x-intercept** and **y-intercept**, we follow these steps:

### 1. Finding the **y-intercept**:
The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation and solve for \( y \):

\[
y = 2 \sqrt[3]{0 - 8} + 2 = 2 \sqrt[3]{-8} + 2 = 2(-2) + 2 = -4 + 2 = -2
\]

Thus, the **y-intercept** is at \( (0, -2) \).

### 2. Finding the **x-intercept**:
The x-intercept occurs when \( y = 0 \). Set \( y = 0 \) and solve for \( x \):

\[
0 = 2 \sqrt[3]{x - 8} + 2
\]

First, subtract 2 from both sides:

\[
-2 = 2 \sqrt[3]{x - 8}
\]

Now, divide by 2:

\[
-1 = \sqrt[3]{x - 8}
\]

Next, cube both sides to eliminate the cube root:

\[
(-1)^3 = x - 8 \implies -1 = x - 8
\]

Solve for \( x \):

\[
x = 7
\]

Thus, the **x-intercept** is at \( (7, 0) \).

### Summary:
- **x-intercept**: \( (7, 0) \)
- **y-intercept**: \( (0, -2) \)

Would you like more details or have any questions about this solution?

Here are 5 related questions you could explore:
1. How would the intercepts change if the function had a different constant term?
2. What is the domain of the function \( y = 2 \sqrt[3]{x - 8} + 2 \)?
3. How do you find the asymptotes for functions with cube roots?
4. Can this function have multiple x-intercepts?
5. How would the graph of the function look, and how would you plot it manually?

**Tip**: When solving for intercepts, always remember that the y-intercept involves setting \( x = 0 \), while the x-intercept involves setting \( y = 0 \).

Find the x and y intercepts of the given function: y = 2∛(x - 8) + 2

Discover how to find the x and y intercepts of the function y = 2∛(x - 8) + 2 with a step-by-step explanation. This example involves solving cube root equations and is suitable for high school students in Grades 10-12.

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