To determine which equation represents a quadratic function with a leading coefficient of 2 and a constant term of -3, let's analyze each option step by step.
1. [tex]\( f(x) = 2x^3 - 3 \)[/tex]
This is a cubic function because the highest power of [tex]\( x \)[/tex] is 3. We need a quadratic function, so this is not the correct choice.
2. [tex]\( f(x) = -3x^2 - 3x + 2 \)[/tex]
This is a quadratic function because the highest power of [tex]\( x \)[/tex] is 2. However, the leading coefficient (coefficient of [tex]\( x^2 \)[/tex]) is -3, not 2. Thus, this is not the correct choice.
3. [tex]\( f(x) = -3x^3 + 2 \)[/tex]
This is a cubic function because the highest power of [tex]\( x \)[/tex] is 3. We need a quadratic function, so this is not the correct choice.
4. [tex]\( f(x) = 2x^2 + 3x - 3 \)[/tex]
This is a quadratic function because the highest power of [tex]\( x \)[/tex] is 2. The leading coefficient (coefficient of [tex]\( x^2 \)[/tex]) is 2, and the constant term is -3. This matches the given requirements perfectly.
Therefore, the equation that represents a quadratic function with a leading coefficient of 2 and a constant term of -3 is:
[tex]\[ \boxed{f(x) = 2x^2 + 3x - 3} \][/tex]
