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You are being asked to graph the function \( f(x) = \sqrt{2x + 1} \). Let me break down how to approach this:

### Key Steps:
1. **Determine the Domain**: 
   - Since the function involves a square root, the expression inside the root must be non-negative:
     \[
     2x + 1 \geq 0 \quad \Rightarrow \quad x \geq -\frac{1}{2}
     \]
   - Therefore, the domain is \( x \geq -\frac{1}{2} \).

2. **Find the Leftmost Point**:
   - The smallest \( x \)-value is \( x = -\frac{1}{2} \), which is where the function starts.
   - At \( x = -\frac{1}{2} \), the function value is:
     \[
     f\left(-\frac{1}{2}\right) = \sqrt{2 \cdot -\frac{1}{2} + 1} = \sqrt{0} = 0
     \]
   - So, the leftmost point is \( \left( -\frac{1}{2}, 0 \right) \).

3. **Choose Additional Points**:
   - Now pick other \( x \)-values greater than or equal to \( -\frac{1}{2} \). Let's try these values:
     - For \( x = 0 \):
       \[
       f(0) = \sqrt{2(0) + 1} = \sqrt{1} = 1 \quad \text{Point: } (0, 1)
       \]
     - For \( x = 1 \):
       \[
       f(1) = \sqrt{2(1) + 1} = \sqrt{3} \approx 1.732 \quad \text{Point: } (1, 1.732)
       \]
     - For \( x = 2 \):
       \[
       f(2) = \sqrt{2(2) + 1} = \sqrt{5} \approx 2.236 \quad \text{Point: } (2, 2.236)
       \]

### Points to Plot:
1. \( \left( -\frac{1}{2}, 0 \right) \)
2. \( (0, 1) \)
3. \( (1, 1.732) \)
4. \( (2, 2.236) \)

You can now plot these points on the graph and connect them to sketch the curve of the function. Let me know if you need further clarification or more points.

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### Questions for Further Exploration:
1. How does the shape of \( f(x) = \sqrt{2x + 1} \) differ from \( f(x) = \sqrt{x} \)?
2. What happens to the graph of \( f(x) = \sqrt{2x + 1} \) if the constant inside the square root changes (e.g., \( \sqrt{2x - 3} \))?
3. What is the range of the function \( f(x) = \sqrt{2x + 1} \)?
4. How would the graph look if the function was \( f(x) = -\sqrt{2x + 1} \)?
5. What is the effect of multiplying the entire function by a constant (e.g., \( f(x) = 3\sqrt{2x + 1} \))?

**Tip**: Always check the domain when dealing with square roots or logarithmic functions.

Graph the function f(x) = √(2x + 1). Plot four points on the graph of the function: the leftmost point and three additional points.

Learn how to graph the square root function f(x) = √(2x + 1). This guide shows how to determine the domain and plot key points, including the leftmost point and three additional points. Suitable for high school students.

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