The graph of which function has an axis of symmetry at $x=-\frac{1}{4}$?
To determine which function has its axis of symmetry at \(x = -\frac{1}{4}\), we recall that the graph of a quadratic function in the standard form \(f(x) = ax^2 + bx + c\) (where \(a
eq 0\)) is a parabola. The axis of symmetry for such a parabola is a vertical line given by the formula:
\[x = -\frac{b}{2a}\]
This formula allows us to identify the axis of symmetry directly from the coefficients of the quadratic equation. Our task is to test each of the given options to see which one results in \(x = -\frac{1}{4}\).
We analyze each function individually by identifying its coefficients \(a\) and \(b\) and calculating the axis of symmetry:
Here, \(a = 2\) and \(b = 1\).
The axis of symmetry is \(x = -\frac{b}{2a} = -\frac{1}{2(2)} = -\frac{1}{4}\).
Here, \(a = 2\) and \(b = -1\).
The axis of symmetry is \(x = -\frac{b}{2a} = -\frac{-1}{2(2)} = \frac{1}{4}\).
Here, \(a = 1\) and \(b = 2\).
The axis of symmetry is \(x = -\frac{b}{2a} = -\frac{2}{2(1)} = -1\).
Here, \(a = 1\) and \(b = -2\).
The axis of symmetry is \(x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1\).
Comparing our results to the required axis of symmetry \(x = -\frac{1}{4}\), we see that only the first function, \(f(x) = 2x^2 + x - 1\), satisfies the condition. Therefore, this is the correct function. The final answer is \(\boxed{f(x) = 2x^2 + x - 1}\).
In architecture, symmetry is often used for aesthetic balance in bridge design. Engineers can use the axis of symmetry of a parabolic arch to ensure that the load distribution is equal on both sides of the center point. For example, if a suspension cable follows the path of \(f(x) = 2x^2 + x - 1\), its lowest point would be located at the axis of symmetry \(x = -\frac{1}{4}\), which helps engineers perfectly align the supports of the bridge. Understanding how to find this point is a fundamental skill in structural engineering.