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 $a$).