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The image contains two mathematical problems for practice. Here's a breakdown of the questions:

### Problem 1: Piecewise Function Evaluation
Evaluate the following piecewise function \( f(x) \) at specified values:
\[ f(x) = \begin{cases} 
\frac{x^2 - 2x}{x + 3} & \text{if } x \geq 3 \\
x^2 + x & \text{if } x < 3 
\end{cases} \]
- (a) \( x = 3 \)
- (b) \( x = 3 \), and \( x = -1 \)

### Problem 2: Graphing a Quadratic Function
Sketch the graph of the function \( f(x) = x^2 + 3x - 5 \).

### Solution Outline:

#### Problem 1:
- **(a) \( x = 3 \):** Since \( x = 3 \geq 3 \), use the first part of the piecewise function:
  \[
  f(3) = \frac{3^2 - 2(3)}{3 + 3} = \frac{9 - 6}{6} = \frac{3}{6} = \frac{1}{2}
  \]
- **(b) \( x = 3 \), and \( x = -1 \):**
  - For \( x = 3 \), as calculated above, \( f(3) = \frac{1}{2} \).
  - For \( x = -1 \), since \( x < 3 \), use the second part of the piecewise function:
    \[
    f(-1) = (-1)^2 + (-1) = 1 - 1 = 0
    \]

#### Problem 2:
To sketch the graph of \( f(x) = x^2 + 3x - 5 \), we need to:
1. Find the vertex using \( x = -\frac{b}{2a} \) where \( a = 1 \) and \( b = 3 \).
2. Calculate the y-intercept by setting \( x = 0 \).
3. Find additional points and symmetry to help sketch the curve.

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### Further Questions to Explore:
1. What are the roots of the quadratic function \( f(x) = x^2 + 3x - 5 \)?
2. How do you determine the vertex of a parabola given by \( f(x) = ax^2 + bx + c \)?
3. What is the significance of the axis of symmetry in a parabola?
4. How does the sign of the leading coefficient affect the direction of a parabola's opening?
5. Can you explain the process of evaluating a piecewise function?
6. How do you determine the domain and range of a piecewise function?
7. What is the importance of continuity when working with piecewise functions?
8. How do you identify key features of a graph from its equation?

### Tip:
When sketching the graph of a quadratic function, it's helpful to plot the vertex and axis of symmetry first, then determine the direction of the parabola's opening based on the leading coefficient.

Explore practice problems involving piecewise function evaluation and graphing quadratic functions. Includes step-by-step solutions for evaluating piecewise functions and sketching the graph of a quadratic function. Suitable for high school students studying advanced algebra and function analysis.

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