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Let's break down the problem you provided.

### 2.1.1 Write down the asymptotes for the graph \( g(x) = -\frac{1}{x+2} - 1 \).

For the function \( g(x) = -\frac{1}{x+2} - 1 \), the asymptotes are:

1. **Vertical Asymptote**: This occurs where the denominator is zero, i.e., where \( x + 2 = 0 \), so:
   \[
   x = -2
   \]
2. **Horizontal Asymptote**: To find the horizontal asymptote, we examine the behavior of \( g(x) \) as \( x \to \infty \) or \( x \to -\infty \). As \( x \) becomes very large (in magnitude), \( -\frac{1}{x+2} \) approaches 0, leaving:
   \[
   y = -1
   \]

So, the **vertical asymptote** is \( x = -2 \) and the **horizontal asymptote** is \( y = -1 \).

---

### 2.1.2 Plot the graphs \( f(x) = -x^2 + 2x + 3 \) and \( g(x) = -\frac{1}{x+2} - 1 \)

Let's analyze the key features required for plotting:

#### For \( f(x) = -x^2 + 2x + 3 \):
- **Intercepts**:
  - **x-intercepts**: Set \( f(x) = 0 \) and solve \( -x^2 + 2x + 3 = 0 \).
  - **y-intercept**: Substitute \( x = 0 \), giving \( f(0) = 3 \).
  
- **Turning Point**: Complete the square or use the formula \( x = \frac{-b}{2a} \) to find the vertex (turning point).

- **Axis of Symmetry**: Given by the \( x \)-coordinate of the turning point.

#### For \( g(x) = -\frac{1}{x+2} - 1 \):
- **Intercepts**:
  - **x-intercept**: Set \( g(x) = 0 \), and solve \( -\frac{1}{x+2} - 1 = 0 \).
  - **y-intercept**: Substitute \( x = 0 \) into \( g(x) \).

- **Asymptotes**: From part 2.1.1, vertical at \( x = -2 \) and horizontal at \( y = -1 \).

Would you like further details on any of these steps or solutions?

### Relative Questions
1. How do you find the turning point of a quadratic function?
2. What are the steps to solve for the x-intercepts of \( f(x) \)?
3. Why does \( g(x) = -\frac{1}{x+2} - 1 \) have a vertical asymptote?
4. How do you determine the horizontal asymptote for rational functions?
5. What is the importance of the axis of symmetry in the graph of a quadratic?

### Tip
Always check the denominator of rational functions for values that make it zero, as these will indicate vertical asymptotes.

Given the graphs, f(x) = -x^2 + 2x + 3 and g(x) = -1/(x + 2) - 1, write down the asymptotes for the graph g and plot both graphs on the same system of axis, showing all the intercepts with the axis, turning point, asymptotes, and axis of symmetry.

Solve the quadratic function f(x) = -x^2 + 2x + 3 and the rational function g(x) = -1/(x + 2) - 1. Learn how to identify asymptotes, turning points, and intercepts for each graph. This problem involves key algebraic concepts and is suitable for students in Grades 10-12.

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