To determine which function satisfies the conditions of being a quadratic function with a leading coefficient of [tex]\( 3 \)[/tex] and a constant term of [tex]\( -12 \)[/tex], we need to analyze each function given.
First, let's recall what these terms mean:
1. Quadratic Function: A function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex].
2. Leading Coefficient: The coefficient [tex]\( a \)[/tex] in front of the [tex]\( x^2 \)[/tex] term.
3. Constant Term: The term [tex]\( c \)[/tex] that does not depend on [tex]\( x \)[/tex].
Now we will analyze each option one by one:
1. [tex]\( f(x) = -12 x^2 + 3 x + 1 \)[/tex]
- Leading coefficient: [tex]\( -12 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex])
- Constant term: [tex]\( 1 \)[/tex]
- This function does not meet our conditions because the leading coefficient is not [tex]\( 3 \)[/tex] and the constant term is not [tex]\( -12 \)[/tex].
2. [tex]\( f(x) = 3 x^2 + 11 x - 12 \)[/tex]
- Leading coefficient: [tex]\( 3 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex])
- Constant term: [tex]\( -12 \)[/tex]
- This function meets both conditions: it is a quadratic function with a leading coefficient of [tex]\( 3 \)[/tex] and a constant term of [tex]\( -12 \)[/tex].
3. [tex]\( f(x) = 12 x^2 + 3 x + 3 \)[/tex]
- Leading coefficient: [tex]\( 12 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex])
- Constant term: [tex]\( 3 \)[/tex]
- This function does not meet our conditions because the leading coefficient is not [tex]\( 3 \)[/tex] and the constant term is not [tex]\( -12 \)[/tex].
4. [tex]\( f(x) = 3 x - 12 \)[/tex]
- Leading coefficient: N/A (It is not a quadratic function because it does not have an [tex]\( x^2 \)[/tex] term)
- Constant term: [tex]\( -12 \)[/tex]
- This function is not a quadratic function, thus it does not meet our conditions.
From the analysis, only option 2) [tex]\( f(x) = 3 x^2 + 11 x - 12 \)[/tex] meets both conditions of being a quadratic function with a leading coefficient of [tex]\( 3 \)[/tex] and a constant term of [tex]\( -12 \)[/tex].
Thus, the correct function is:
[tex]\[ f(x) = 3 x^2 + 11 x - 12 \][/tex]
