The axis of symmetry for a function in the form $f(x)=x^2+4 x-5$ is $x=-2$. What are the coordinates of the vertex of the graph?
The given quadratic function is \(f(x) = x^2 + 4x - 5\). The axis of symmetry for any parabola defined by a quadratic function \(f(x) = ax^2 + bx + c\) is a vertical line passing through its vertex. Since the axis of symmetry is given as \(x = -2\), the x-coordinate of the vertex, denoted as \(h\), must be \(-2\).
The y-coordinate of the vertex, denoted as \(k\), is found by evaluating the function at the x-coordinate \(h\). We substitute \(x = -2\) into the function \(f(x) = x^2 + 4x - 5\):
\[k = f(-2) = (-2)^2 + 4(-2) - 5\]
Calculating each term:
\[(-2)^2 = 4\]
\[4(-2) = -8\]
So,
\[k = 4 - 8 - 5\]
\[k = -4 - 5\]
\[k = -9\]
Having found \(h = -2\) and \(k = -9\), the coordinates of the vertex \((h, k)\) are \((-2, -9)\). Comparing this result with the given choices, it matches option (b). The vertex of the graph of \(f(x) = x^2 + 4x - 5\) is \(\boxed{(-2, -9)}\).
Understanding the vertex of a parabola is essential in fields like physics and engineering, particularly when modeling projectile motion. For example, if a ball is thrown into the air, its height as a function of time can be modeled by a quadratic equation. The vertex of this parabola represents the highest point the ball reaches (the maximum height) and the time at which this occurs. By finding the vertex coordinates, one can easily determine the maximum performance or safety limits in various real-world scenarios, such as designing sports equipment or optimizing mechanical trajectories.