To determine which quadratic function in standard form has the given values [tex]\(a = -3.5\)[/tex], [tex]\(b = 2.7\)[/tex], and [tex]\(c = -8.2\)[/tex], we need to carefully compare these values with each function provided. The standard form of a quadratic function is given by:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
Let’s go through each option one by one:
1. [tex]\(f(x) = 2.7 x^2 - 8.2 x - 3.5\)[/tex]:
- Here, comparing coefficients:
- [tex]\(a = 2.7\)[/tex]
- [tex]\(b = -8.2\)[/tex]
- [tex]\(c = -3.5\)[/tex]
- This does not match our given values of [tex]\(a = -3.5\)[/tex], [tex]\(b = 2.7\)[/tex], and [tex]\(c = -8.2\)[/tex].
2. [tex]\(f(x) = 2.7 x^2 - 3.5 x - 8.2\)[/tex]:
- Here, comparing coefficients:
- [tex]\(a = 2.7\)[/tex]
- [tex]\(b = -3.5\)[/tex]
- [tex]\(c = -8.2\)[/tex]
- This does not match our given values either.
3. [tex]\(f(x) = -3.5 x^2 - 8.2 x + 2.7\)[/tex]:
- Here, comparing coefficients:
- [tex]\(a = -3.5\)[/tex]
- [tex]\(b = -8.2\)[/tex]
- [tex]\(c = 2.7\)[/tex]
- This also does not match our given values.
4. [tex]\(f(x) = -3.5 x^2 + 2.7 x - 8.2\)[/tex]:
- Here, comparing coefficients:
- [tex]\(a = -3.5\)[/tex]
- [tex]\(b = 2.7\)[/tex]
- [tex]\(c = -8.2\)[/tex]
- This matches our given values of [tex]\(a = -3.5\)[/tex], [tex]\(b = 2.7\)[/tex], and [tex]\(c = -8.2\)[/tex].
From this detailed comparison, we can conclude that the correct quadratic function with the values [tex]\(a = -3.5\)[/tex], [tex]\(b = 2.7\)[/tex], and [tex]\(c = -8.2\)[/tex] is:
[tex]\[ \boxed{f(x) = -3.5 x^2 + 2.7 x - 8.2} \][/tex]
