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.