https://file.aiquickdraw.com/mathbot/file/1730396010882dleh8mqm.jpg

Let's address each question one by one:

### Question 1
The function is given as:
\[
f(x) = 2(x - 5)^2 + 12
\]
This is in **vertex form** for a parabola, \( f(x) = a(x - h)^2 + k \), where the vertex is at \( (h, k) \).

From this form:
- \( h = 5 \)
- \( k = 12 \)

Thus, the vertex of \( f(x) \) is:
\[
(5, 12)
\]

### Question 2
The function is given as:
\[
f(x) = (x + 7)(x - 3)
\]
To convert this to standard form, we need to expand it:

1. Multiply each term:
   \[
   f(x) = x \cdot x + x \cdot (-3) + 7 \cdot x + 7 \cdot (-3)
   \]
2. Simplify:
   \[
   f(x) = x^2 - 3x + 7x - 21
   \]
3. Combine like terms:
   \[
   f(x) = x^2 + 4x - 21
   \]

So, the standard form is:
\[
f(x) = x^2 + 4x - 21
\]
This matches option **C**.

---

Would you like further clarification on any steps, or have any additional questions?

### Related Questions
1. What is the significance of the vertex form of a quadratic function?
2. How can the standard form of a quadratic equation help determine the direction of a parabola?
3. How does changing the \( a \) value in vertex form affect the parabola's width?
4. Can the vertex form of a quadratic function be derived from its standard form?
5. What is the process of completing the square, and how does it relate to converting standard form to vertex form?

### Tip
Remember, in the vertex form \( f(x) = a(x - h)^2 + k \), the vertex is always \( (h, k) \) regardless of the value of \( a \). The sign of \( a \) determines if the parabola opens upwards (positive \( a \)) or downwards (negative \( a \)).

What is the vertex of f(x) = 2(x - 5)^2 + 12 and Convert (x + 7)(x - 3) to Standard Form

Learn how to find the vertex of a quadratic function given in vertex form and how to convert a factored form expression into standard form. This problem covers essential concepts in Algebra, including the vertex and standard forms of quadratic functions. Ideal for students in Grades 9-10.

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