To determine which quadratic function has a leading coefficient of 3 and a constant term of -12, we need to:
1. Identify the leading coefficient of each quadratic function.
2. Identify the constant term of each quadratic function.
3. Check which function satisfies both conditions: having a leading coefficient of 3 and a constant term of -12.
Let's analyze each given function:
- For [tex]\( f(x) = -12x^2 + 3x + 1 \)[/tex]:
- The leading coefficient is -12.
- The constant term is 1.
- This does not meet the criteria.
- For [tex]\( f(x) = 3x^2 + 11x - 12 \)[/tex]:
- The leading coefficient is 3.
- The constant term is -12.
- This meets the criteria.
- For [tex]\( f(x) = 12x^2 + 3x + 3 \)[/tex]:
- The leading coefficient is 12.
- The constant term is 3.
- This does not meet the criteria.
- For [tex]\( f(x) = 3x - 12 \)[/tex]:
- This is not a quadratic function; it is a linear function.
- Hence, it does not meet the criteria.
After evaluating all the given options, the quadratic function that has a leading coefficient of 3 and a constant term of -12 is:
[tex]\[ f(x) = 3x^2 + 11x - 12 \][/tex]
Thus, the correct answer is:
[tex]\[ f(x) = 3x^2 + 11x - 12 \][/tex]
