https://file.aiquickdraw.com/mathbot/file/1729100924954kbv2ugmi.jpg

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!

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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.

Given the quadratic function in vertex form: f(x) = (x + 1)^2 - 2.

a. Determine the coordinates of the vertex.
b. Write the quadratic in standard form.

Learn how to find the vertex and convert the quadratic function f(x) = (x + 1)^2 - 2 from vertex form to standard form. This problem covers key algebraic concepts and steps for expanding quadratic functions, suitable for Grades 9-11.

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