To find \( f(1) \), we need to rewrite the function \( f(x+3) \) in terms of \( f(x) \).

Given:
\[
f(x + 3) = x^2 + 8x + 12
\]

Let \( y = x + 3 \). Then:
\[
f(y) = (y - 3)^2 + 8(y - 3) + 12
\]

Now, substitute \( y = 1 \) into this expression to find \( f(1) \).

If \( y = 1 \), then \( x = y - 3 = 1 - 3 = -2 \).

Thus, substitute \( x = -2 \) into the expression \( f(x+3) = x^2 + 8x + 12 \) to find the result:
\[
f(1) = (-2)^2 + 8(-2) + 12
\]

Now calculate:
\[
f(1) = 4 - 16 + 12 = 0
\]

So, \( f(1) = 0 \).

Would you like further details on any steps or have any questions?

Here are five related questions you might find helpful:
1. How would you find \( f(2) \) using the same function?
2. What is the general process for shifting functions horizontally?
3. How can you derive the vertex form of a quadratic function from its standard form?
4. What are the roots of the quadratic equation \( x^2 + 8x + 12 = 0 \)?
5. How would the graph of \( f(x) \) look if you were to sketch it?

**Tip**: When solving function transformations, always identify how the variable is shifted (e.g., \( x + 3 \)) to simplify calculations.

Explore how to find f(1) for a function transformed by shifting. Learn step-by-step with examples and formulas. Suitable for high school students studying function transformations and quadratic equations.

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