The quadratic function given is in vertex form:

$f(x) = (x + 1)^2 - 2$

Part a: Determine the coordinates of the vertex.

The vertex form of a quadratic function is generally written as:

$f(x) = a(x - h)^2 + k$

Where $(h, k)$ are the coordinates of the vertex. Comparing the given function with the standard form, we can see that:

• $h = -1$ (because of $(x + 1)$)
• $k = -2$

So, the vertex is:

$\boxed{(-1, -2)}$

Part b: Write the quadratic in standard form.

To convert the function to standard form, expand $(x + 1)^2 - 2$:

$(x + 1)^2 = x^2 + 2x + 1$

So,

$f(x) = x^2 + 2x + 1 - 2 = x^2 + 2x - 1$

Thus, the quadratic in standard form is:

$\boxed{f(x) = x^2 + 2x - 1}$

Let me know if you want more details or explanations!

Here are 5 related questions to expand your understanding:

1. What is the significance of the vertex form in identifying the vertex of a parabola?
2. How does the sign of $a$ in the vertex form affect the direction of the parabola?
3. Can you derive the axis of symmetry from the vertex form?
4. How do you complete the square to transform a quadratic from standard form to vertex form?
5. What is the importance of the discriminant in determining the nature of the roots of a quadratic equation?

Tip: Always remember that the vertex of a quadratic in vertex form $f(x) = a(x-h)^2 + k$ is simply $(h, k)$, making it easier to locate the maximum or minimum point of the parabola.