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The question asks which of the following functions are equivalent to \( f(x) = 3x^2 + 6x - 45 \). Let's analyze the options to determine equivalence by simplifying each one:

### Given function:
\[ f(x) = 3x^2 + 6x - 45 \]

### Option 1: 
\[ f(x) = 3(x + 1)^2 - 48 \]

Let's expand this:
\[
(x + 1)^2 = x^2 + 2x + 1
\]
Now multiply by 3:
\[
3(x^2 + 2x + 1) = 3x^2 + 6x + 3
\]
Subtract 48:
\[
3x^2 + 6x + 3 - 48 = 3x^2 + 6x - 45
\]
This matches the original function, so **Option 1 is equivalent**.

### Option 2:
\[ f(x) = 3(x + 15)(x - 3) \]

Let's expand this:
\[
(x + 15)(x - 3) = x^2 - 3x + 15x - 45 = x^2 + 12x - 45
\]
Now multiply by 3:
\[
3(x^2 + 12x - 45) = 3x^2 + 36x - 135
\]
This does **not** match the original function, so **Option 2 is not equivalent**.

### Option 3:
\[ f(x) = 3(x - 3)(x + 5) \]

Let's expand this:
\[
(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15
\]
Now multiply by 3:
\[
3(x^2 + 2x - 15) = 3x^2 + 6x - 45
\]
This matches the original function, so **Option 3 is equivalent**.

### Option 4:
\[ f(x) = 3(3x + 3)^2 - 45 \]

Let's expand this:
\[
(3x + 3)^2 = 9x^2 + 18x + 9
\]
Now multiply by 3:
\[
3(9x^2 + 18x + 9) = 27x^2 + 54x + 27
\]
Subtract 45:
\[
27x^2 + 54x + 27 - 45 = 27x^2 + 54x - 18
\]
This does **not** match the original function, so **Option 4 is not equivalent**.

### Conclusion:
The equivalent functions are **Option 1** and **Option 3**.

Let me know if you'd like more details or have any questions! Here are 5 related questions:

1. What is the process for factoring quadratic equations like \( 3x^2 + 6x - 45 \)?
2. How can we complete the square to rewrite quadratic functions?
3. How does expanding binomials work, and how can it help solve equivalent function problems?
4. How do you graph quadratic functions and analyze their transformations?
5. What are other methods for determining if two functions are equivalent?

**Tip:** When checking for equivalence between functions, always expand and simplify to compare their terms directly.

Which of the following functions are equivalent to f(x) = 3x^2 + 6x - 45?

Explore the process of determining function equivalence for quadratic expressions. Learn how to expand binomials and simplify quadratic functions like f(x) = 3x^2 + 6x - 45. This guide is suitable for students in Grades 9-11, focusing on algebra and function transformations.

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