What is the axis of symmetry for the function $f(x)=7-4 x+x^2$?
The given quadratic function is \(f(x) = 7 - 4x + x^2\). We can rewrite this function in the standard form \(f(x) = ax^2 + bx + c\) by reordering the terms:
\[f(x) = x^2 - 4x + 7\]
From this form, we can identify the coefficients:
The axis of symmetry for a parabola defined by a quadratic function \(f(x) = ax^2 + bx + c\) is a vertical line that passes through the vertex of the parabola. The formula to find the \(x\)-coordinate of the axis of symmetry is given by:
\[x = -\frac{b}{2a}\]
This formula is derived from the vertex form of the quadratic equation.
Using the coefficients identified in the first step (\(a = 1\) and \(b = -4\)), we substitute these values into the axis of symmetry formula:
\[x = -\frac{-4}{2(1)}\]
Simplifying the expression:
\[x = -\left(\frac{-4}{2}\right)\]
\[x = -(-2)\]
\[x = 2\]
Thus, the axis of symmetry for the given function is the vertical line \(x = 2\). Comparing this result with the provided options:
The correct option is b.
The concept of the axis of symmetry is essential in engineering and architecture. For example, when designing a parabolic arch for a bridge or a reflective satellite dish, the axis of symmetry ensures that the design is balanced and distributes stress evenly. By understanding where the axis of symmetry lies, engineers can precisely place support structures or focus sensors at the optimal point, creating stable and efficient designs.