To determine which of the given functions are quadratic functions, we should first understand the definition of a quadratic function. A quadratic function is a polynomial function of degree 2, which means the highest power of [tex]\( x \)[/tex] in the function is [tex]\( x^2 \)[/tex] and there are no terms with a higher power than [tex]\( x^2 \)[/tex].
Let's analyze each of the given functions:
a. [tex]\( f(x) = x^2 - x \)[/tex]
- The highest power of [tex]\( x \)[/tex] in this function is [tex]\( x^2 \)[/tex].
- There are no terms with a higher power than [tex]\( x^2 \)[/tex].
- Therefore, this is a quadratic function.
b. [tex]\( f(x) = x + 2 \)[/tex]
- The highest power of [tex]\( x \)[/tex] in this function is [tex]\( x \)[/tex].
- There is no [tex]\( x^2 \)[/tex] term.
- Therefore, this is not a quadratic function.
c. [tex]\( f(x) = 2x^2 - 9x + 3 \)[/tex]
- The highest power of [tex]\( x \)[/tex] in this function is [tex]\( x^2 \)[/tex].
- There are no terms with a higher power than [tex]\( x^2 \)[/tex].
- Therefore, this is a quadratic function.
d. [tex]\( f(x) = x^3 - 2x^2 + x \)[/tex]
- The highest power of [tex]\( x \)[/tex] in this function is [tex]\( x^3 \)[/tex].
- Since the highest power is [tex]\( x^3 \)[/tex], this is not a quadratic function.
Thus, the functions that are quadratic are:
- Function a: [tex]\( f(x) = x^2 - x \)[/tex]
- Function c: [tex]\( f(x) = 2x^2 - 9x + 3 \)[/tex]
Therefore, the correct answers are:
- a. [tex]\( f(x) = x^2 - x \)[/tex]
- c. [tex]\( f(x) = 2x^2 - 9x + 3 \)[/tex]
The indices of the quadratic functions are:
[tex]\[ \boxed{[1, 3]} \][/tex]
