To determine which function has an axis of symmetry at [tex]\(x = -\frac{1}{4}\)[/tex], we need to understand the concept of the axis of symmetry for a quadratic function.
For a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex], the formula for the axis of symmetry is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
We will apply this formula to each given function and determine which one results in [tex]\( x = -\frac{1}{4} \)[/tex].
### 1. Function: [tex]\( f(x) = 2x^2 + x - 1 \)[/tex]
Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = 1 \)[/tex].
[tex]\[ x = -\frac{b}{2a} = -\frac{1}{2 \times 2} = -\frac{1}{4} \][/tex]
This matches the given axis of symmetry [tex]\( x = -\frac{1}{4} \)[/tex].
### 2. Function: [tex]\( f(x) = 2x^2 - x + 1 \)[/tex]
Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = -1 \)[/tex].
[tex]\[ x = -\frac{b}{2a} = -\frac{-1}{2 \times 2} = \frac{1}{4} \][/tex]
This does not match the given axis of symmetry [tex]\( x = -\frac{1}{4} \)[/tex].
### 3. Function: [tex]\( f(x) = x^2 + 2x - 1 \)[/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex].
[tex]\[ x = -\frac{b}{2a} = -\frac{2}{2 \times 1} = -1 \][/tex]
This does not match the given axis of symmetry [tex]\( x = -\frac{1}{4} \)[/tex].
### 4. Function: [tex]\( f(x) = x^2 - 2x + 1 \)[/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -2 \)[/tex].
[tex]\[ x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1 \][/tex]
This does not match the given axis of symmetry [tex]\( x = -\frac{1}{4} \)[/tex].
### Conclusion
The function whose graph has an axis of symmetry at [tex]\( x = -\frac{1}{4} \)[/tex] is:
[tex]\[ \boxed{f(x) = 2x^2 + x - 1} \][/tex]
