Determine the x- and y-intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure 16(a). In Exercises 31-34, you will first need to factor the polynomial. Y=(x-2)(x-1)(x+1)

To solve the problem, let's determine the x-intercepts, y-intercepts, and excluded regions for the given function \( y = (x-2)(x-1)(x+1) \). I'll also explain how you can sketch the graph.

### 1. **X-Intercepts**
The x-intercepts occur where \( y = 0 \). Since \( y = (x-2)(x-1)(x+1) \), the x-intercepts can be found by setting each factor equal to zero:

\[
(x-2) = 0 \quad \Rightarrow \quad x = 2
\]
\[
(x-1) = 0 \quad \Rightarrow \quad x = 1
\]
\[
(x+1) = 0 \quad \Rightarrow \quad x = -1
\]

So, the x-intercepts are \( x = 2 \), \( x = 1 \), and \( x = -1 \). These correspond to the points \( (2, 0) \), \( (1, 0) \), and \( (-1, 0) \).

### 2. **Y-Intercept**
The y-intercept occurs where \( x = 0 \). Plugging \( x = 0 \) into the equation:

\[
y = (0-2)(0-1)(0+1) = (-2)(-1)(1) = 2
\]

So, the y-intercept is \( (0, 2) \).

### 3. **Excluded Regions**
For polynomial functions, there are no excluded regions as polynomials are defined for all real numbers. Therefore, there are no values of \( x \) that would make the function undefined.

### 4. **Sketching the Graph**
- **X-intercepts:** Plot the points \( (2, 0) \), \( (1, 0) \), and \( (-1, 0) \) on the x-axis.
- **Y-intercept:** Plot the point \( (0, 2) \) on the y-axis.
- **Graph Shape:** The function \( y = (x-2)(x-1)(x+1) \) is a cubic polynomial, which means the graph will cross the x-axis at each of the x-intercepts and have a general "S" shape.

Here's a description of the graph:
- The graph starts from negative infinity, rises through \( (-1, 0) \), dips down, and rises again through \( (1, 0) \), then crosses the y-axis at \( (0, 2) \), and finally crosses the x-axis again at \( (2, 0) \).
  
If you want, I can generate a sketch of this graph for you.

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Would you like to see a visual sketch of this graph? Let me know if you need any more details or if you have any questions!

### Follow-up Questions:
1. What is the significance of the x-intercepts in this function?
2. How does the degree of the polynomial influence the shape of the graph?
3. Can you identify the turning points of the graph?
4. What would happen if one of the roots was repeated (i.e., if the function had a factor like \( (x-2)^2 \))?
5. How would the graph change if the function had a negative leading coefficient?

**Tip:** When sketching polynomials, always consider the degree and the sign of the leading coefficient to predict the general shape of the graph.

Discover how to determine the x- and y-intercepts of the polynomial function Y=(x-2)(x-1)(x+1) and understand how to sketch its graph. This exercise covers polynomial functions, factoring, and graphing techniques. Perfect for students in Grades 10-12.

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