\( f ( x ) = x ^ { 2 } + 4 x + 1 , \quad c = 1 \)

Did you know that the function \( f(x) = x^2 + 4x + 1 \) is a quadratic function, and its graph is a parabola? The vertex of this parabola can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 4 \). By calculating this, we find the vertex lies at \( x = -2 \), making it a great place to understand the function's minimum value! In real-world applications, quadratics like this can model various situations such as projectile motion or optimizing area. For instance, if you were designing a garden, you could use a quadratic model to determine the maximum area you can achieve with a given perimeter. It's a practical tool in fields like architecture and environmental design!