To determine which given function is a quadratic function with a leading coefficient of 3 and a constant term of -12, let's analyze each option step-by-step:
1. Analyze the first function:
[tex]\( f(x) = -12x^2 + 3x + 1 \)[/tex]
- The leading coefficient is the coefficient of [tex]\(x^2\)[/tex], which is [tex]\(-12\)[/tex].
- The constant term is [tex]\(1\)[/tex].
This does not match the requirements of a leading coefficient of 3 and a constant term of -12.
2. Analyze the second function:
[tex]\( f(x) = 3x^2 + 11x - 12 \)[/tex]
- The leading coefficient is the coefficient of [tex]\(x^2\)[/tex], which is [tex]\(3\)[/tex].
- The constant term is [tex]\(-12\)[/tex].
This correctly matches the requirements of a leading coefficient of 3 and a constant term of -12.
3. Analyze the third function:
[tex]\( f(x) = 12x^2 + 3x + 3 \)[/tex]
- The leading coefficient is the coefficient of [tex]\(x^2\)[/tex], which is [tex]\(12\)[/tex].
- The constant term is [tex]\(3\)[/tex].
This does not match the requirements of a leading coefficient of 3 and a constant term of -12.
4. Analyze the fourth function:
[tex]\( f(x) = 3x - 12 \)[/tex]
- This is not a quadratic function; it is a linear function since there is no [tex]\(x^2\)[/tex] term.
- The highest power of [tex]\(x\)[/tex] in this function is 1.
Hence, this doesn't qualify as it lacks the [tex]\(x^2\)[/tex] term entirely.
After analyzing each option, the second function [tex]\( f(x) = 3x^2 + 11x - 12 \)[/tex] is the only one that meets the criteria of having a leading coefficient of 3 and a constant term of -12.
Thus, the correct function is:
[tex]\[ f(x)=3x^2+11x-12 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
