To determine which quadratic function in standard form matches the given values \( a = -3.5 \), \( b = 2.7 \), and \( c = -8.2 \), let’s recall the general structure of a quadratic function. A quadratic function in its standard form is expressed as:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
We need to identify which of the provided functions fits this form with the specific coefficients \( a \), \( b \), and \( c \).
Let's go through each option to see which one matches:
1. Option 1: \( f(x) = 2.7x^2 - 8.2x - 3.5 \)
- Here, the coefficients are:
- \( a \) (coefficient of \( x^2 \)) is \( 2.7 \)
- \( b \) (coefficient of \( x \)) is \( -8.2 \)
- \( c \) (constant term) is \( -3.5 \)
- Compare with given values:
- \( a = 2.7 \) (incorrect, should be \( -3.5 \))
- \( b = -8.2 \) (incorrect, should be \( 2.7 \))
- \( c = -3.5 \) (one correct, but overall incorrect match)
2. Option 2: \( f(x) = 2.7x^2 - 3.5x - 8.2 \)
- Here, the coefficients are:
- \( a \) is \( 2.7 \)
- \( b \) is \( -3.5 \)
- \( c \) is \( -8.2 \)
- Compare with given values:
- \( a = 2.7 \) (incorrect, should be \( -3.5 \))
- \( b = -3.5 \) (incorrect, should be \( 2.7 \))
- \( c = -8.2 \) (one correct, but overall incorrect match)
3. Option 3: \( f(x) = -3.5x^2 - 8.2x + 2.7 \)
- Here, the coefficients are:
- \( a \) is \( -3.5 \)
- \( b \) is \( -8.2 \)
- \( c \) is \( 2.7 \)
- Compare with given values:
- \( a = -3.5 \) (correct)
- \( b = -8.2 \) (incorrect, should be \( 2.7 \))
- \( c = 2.7 \) (incorrect, should be \( -8.2 \))
- Overall incorrect match
4. Option 4: \( f(x) = -3.5x^2 + 2.7x - 8.2 \)
- Here, the coefficients are:
- \( a \) is \( -3.5 \)
- \( b \) is \( 2.7 \)
- \( c \) is \( -8.2 \)
- Compare with given values:
- \( a = -3.5 \) (correct)
- \( b = 2.7 \) (correct)
- \( c = -8.2 \) (correct)
- Overall correct match
Therefore, the correct quadratic function in standard form with \( a = -3.5 \), \( b = 2.7 \), and \( c = -8.2 \) is:
[tex]\[ f(x) = -3.5x^2 + 2.7x - 8.2 \][/tex]
Hence, the answer is:
[tex]\[ \boxed{4} \][/tex]
