Question

which equation represents a quadratic function with a leading coefficient of 2 and a constant term of -3?\
\\( f(x) = 2x^3 - 3 \\)\
\\( f(x) = -3x^2 - 3x + 2 \\)\
\\( f(x) = -3x^3 + 2 \\)\
\\( f(x) = 2x^2 + 3x - 3 \\)

Explanation:

Step1: Recall quadratic function form

A quadratic function has the form \( f(x) = ax^2 + bx + c \), where \( a \) is the leading coefficient, \( b \) is the linear coefficient, and \( c \) is the constant term. The degree (highest power of \( x \)) must be 2.

Step2: Analyze each option

• Option 1: \( f(x) = 2x^3 - 3 \) has degree 3 (cubic), not quadratic. Eliminate.
• Option 2: \( f(x) = -3x^2 - 3x + 2 \) has leading coefficient -3 (not 2) and constant term 2 (not -3). Eliminate.
• Option 3: \( f(x) = -3x^3 + 2 \) has degree 3 (cubic), not quadratic. Eliminate.
• Option 4: \( f(x) = 2x^2 + 3x - 3 \) has degree 2 (quadratic), leading coefficient \( a = 2 \), and constant term \( c = -3 \). This matches.

Answer:

\( f(x) = 2x^2 + 3x - 3 \) (the fourth option)