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Quadratic functions are a type of polynomial function where the highest degree of the variable is squared, commonly written in the form \( ax^2 + bx + c \). In these functions:

• \(a\) is the coefficient of the squared term.
• \(b\) is the coefficient of the linear term.
• \(c\) is the constant term or y-intercept.

Understanding quadratic functions is important because they model many real-world scenarios, like the path of an object in free fall.For instance, in our given function \(f(x) = x^2 + 4x\), we can identify \(a = 1\), \(b = 4\), and \(c = 0\). This means it's a basic quadratic function without a vertical shift. A key feature of quadratic functions is their graph, which is always a parabola. Depending on the sign of \(a\), the parabola opens up (\(a > 0\)) or down (\(a < 0\)). In this case, since \(a = 1\), our parabola opens upwards.

Graphing functions visually represents the relationship described by a function, offering an instant understanding of its behavior.To graph a quadratic function like \(f(x) = x^2 + 4x\) for \(x \geq -2\), you start by identifying its vertex, the turning point of the parabola. The formula for the vertex of a parabola \(y = ax^2 + bx + c\) is provided by \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). For our function:

• \(x\)-coordinate of the vertex: \(-\frac{4}{2 \times 1} = -2\)
• \(y\)-coordinate: \(f(-2) = (-2)^2 + 4 \times (-2) = -4\)

Thus, the vertex is at \((-2, -4)\).From here, plot the vertex and some additional points by choosing \(x\) values greater than or equal to -2, to observe the curve's shape. Similarly, when graphing the inverse function \(f^{-1}(x) = \frac{-4 + \sqrt{x + 4}}{2}\), plot points starting from the vertex of the original function, noting that the inverse may be more like half of a parabola on its side.

The quadratic formula is vital for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It is expressed as:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the roots of any quadratic equation, which are the values of \(x\) for which the function equals zero. The term under the square root \((b^2 - 4ac)\) is called the discriminant. This discriminant determines:

• If \(b^2 - 4ac > 0\), the equation has two real and distinct solutions.
• If \(b^2 - 4ac = 0\), there is one real, repeated solution.
• If \(b^2 - 4ac < 0\), the solutions are complex and not real.

For our inverse problem, the quadratic formula helps in solving for \(y\), now rewritten due to swapping roles with \(x\). Using \(a = 1\), \(b = 4\), and \(c = -x\), we apply it to derive the inverse \(f^{-1}(x)\). Solving using this formula ensures accurate determination of the inverse function, crucial for correctly mapping our graph and understanding the transition between functions.