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Let's break down the given function \( f(x) = 2(x + 3)^2 - 1 \) to extract the key information needed for graphing and analyzing it:

### 1. **Vertex of the Parabola**
The equation is in **vertex form**: 
\[
f(x) = a(x-h)^2 + k
\]
Here, \( a = 2 \), \( h = -3 \), and \( k = -1 \). 
Thus, the **vertex** of the parabola is at \( (-3, -1) \).

### 2. **Axis of Symmetry**
The axis of symmetry is the vertical line that passes through the vertex. Hence, the equation of the axis of symmetry is:
\[
x = -3
\]

### 3. **Y-Intercept**
To find the y-intercept, set \( x = 0 \):
\[
f(0) = 2(0 + 3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17
\]
Thus, the y-intercept is \( (0, 17) \).

### 4. **X-Intercepts**
To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[
2(x + 3)^2 - 1 = 0
\]
\[
2(x + 3)^2 = 1
\]
\[
(x + 3)^2 = \frac{1}{2}
\]
\[
x + 3 = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2}
\]
\[
x = -3 \pm \frac{\sqrt{2}}{2}
\]
The two x-intercepts are approximately:
\[
x_1 = -3 + \frac{\sqrt{2}}{2} \approx -2.29 \quad \text{and} \quad x_2 = -3 - \frac{\sqrt{2}}{2} \approx -3.71
\]

### 5. **Domain and Range**
- **Domain**: Since this is a parabola, the domain is all real numbers: 
  \[
  (-\infty, \infty)
  \]
- **Range**: The parabola opens upwards (since \( a = 2 > 0 \)), and the minimum value occurs at the vertex \( (-3, -1) \). Therefore, the range is:
  \[
  [-1, \infty)
  \]

### Graphing the Parabola:
- The vertex is at \( (-3, -1) \).
- The y-intercept is at \( (0, 17) \).
- The x-intercepts are approximately \( (-2.29, 0) \) and \( (-3.71, 0) \).
- The axis of symmetry is \( x = -3 \).

Would you like further details on any specific part of the solution, or should I help with plotting the graph in more detail?

Here are some additional related questions you can explore:
1. How does the value of \( a \) affect the width of the parabola?
2. What would happen if \( h \) or \( k \) were changed in the vertex form equation?
3. How can you verify the x-intercepts algebraically?
4. What are other forms of quadratic functions, and how do they differ?
5. What tools can help you graph this parabola manually or digitally?

**Tip**: When graphing quadratic functions, always start with the vertex and axis of symmetry, then use additional points like intercepts to get an accurate shape.

Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range.

Learn how to graph a quadratic function given its vertex and intercepts. This problem explores the axis of symmetry, domain, and range of the function, along with key graphing techniques for high school students.

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