The given function is \( f(x+h) = 3x^2 + 4x - 3 \).

To find the original function \( f(x) \), let's compare it to the general form of the function \( f(x+h) \), which is usually \( f(x+h) = f(x) \) with \( x \) replaced by \( x+h \).

Here, let's see how the function \( f(x) \) is structured:

1. Start with \( f(x+h) = 3(x+h)^2 + 4(x+h) - 3 \).
2. Expand the expression:

\[
f(x+h) = 3(x^2 + 2xh + h^2) + 4x + 4h - 3
\]
\[
= 3x^2 + 6xh + 3h^2 + 4x + 4h - 3
\]

Now, compare this with the given \( f(x+h) = 3x^2 + 4x - 3 \).

For the expressions to be identical:

- The term \( 6xh + 4h \) must be zero, so \( h \) must be zero.
- The term \( 3h^2 \) must also be zero, which implies \( h = 0 \).

Therefore, the original function is:

\[
f(x) = 3x^2 + 4x - 3
\]

### Questions:
1. What is the derivative of \( f(x) = 3x^2 + 4x - 3 \)?
2. How do you find \( f'(x) \) using the difference quotient method?
3. What is the vertex of the parabola described by \( f(x) = 3x^2 + 4x - 3 \)?
4. How do you find the roots of \( f(x) = 3x^2 + 4x - 3 \)?
5. What does the function \( f(x) = 3x^2 + 4x - 3 \) represent graphically?

### Tip:
To find the vertex of a quadratic function \( ax^2 + bx + c \), use the formula \( x = -\frac{b}{2a} \) for the x-coordinate of the vertex.

Explore how to determine the original function f(x) from the transformed function f(x+h) = 3x^2 + 4x - 3. This problem delves into key algebraic concepts, including quadratic functions and function transformations, and is suitable for students in Grades 9-12.

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