https://file.aiquickdraw.com/mathbot/file/1729469953609xdz90fbe.jpg

The function provided in the question is:

\[
f(x) = x^3 - 4x^2 - x + 4
\]

Let's go over the questions one by one.

1. **At which x-intercept(s) does the graph cross the x-axis?**
   - From the image, the correct answer selected is **\( x = 1, 4, -1 \)**. 
   - This means the graph crosses the x-axis at these values of \( x \).

2. **At which x-intercept(s) does the graph touch the x-axis and turn around?**
   - The answer chosen is **"There are no x-intercepts at which the graph touches the x-axis and turns around."**
   - This is correct because in a cubic function, like the one given, unless there is a double root, the graph usually crosses the x-axis without turning around.

3. **Find the y-intercept.**
   - The y-intercept occurs when \( x = 0 \). Plugging \( x = 0 \) into \( f(x) \):
   \[
   f(0) = 0^3 - 4(0)^2 - 0 + 4 = 4
   \]
   - The y-intercept is \( y = 4 \), which matches the answer provided in the image.

4. **Determine whether the graph has y-axis symmetry, origin symmetry, or neither.**
   - The selected answer is currently blank.
   - To check for symmetry:
     - **Y-axis symmetry**: This happens if \( f(-x) = f(x) \), but here \( f(-x) \neq f(x) \), so it doesn’t have y-axis symmetry.
     - **Origin symmetry**: This happens if \( f(-x) = -f(x) \). Checking this, we can see it doesn't hold either.
   - Therefore, the correct answer is **"neither"**.

Would you like more details on any specific part?

### Here are 5 related questions:
1. How do you find the x-intercepts of a polynomial function?
2. What does it mean if a graph "turns around" at an intercept?
3. How can you algebraically check if a graph has symmetry?
4. How does the degree of a polynomial affect the shape of its graph?
5. Why is the y-intercept always found when \( x = 0 \)?

### Tip:
For polynomials, the sign of the leading coefficient and the degree can give you an idea of the end behavior of the graph.

f(x) = x^3 - 4x^2 - x + 4. At which x-intercept(s) does the graph cross the x-axis? Where does it touch and turn around? Find the y-intercept. Determine symmetry (y-axis, origin, or neither).

Solve the cubic polynomial f(x) = x^3 - 4x^2 - x + 4 by finding its x- and y-intercepts and determining its symmetry. Learn key algebraic concepts like polynomial functions and intercepts, suitable for Grades 10-12.

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