Enter the coordinates of the vertex of the graph of $y=2(x+5)^2-4$.
Vertex: ( □ , □ )
The given function is \(y = 2(x + 5)^2 - 4\). This is a quadratic function expressed in vertex form. The vertex form of a parabola is defined by the equation \(y = a(x - h)^2 + k\), where \((h, k)\) represents the coordinates of the vertex of the parabola. Identifying the vertex allows us to understand the peak or the turning point of the parabolic curve.
To identify the vertex \((h, k)\), we compare the provided equation \(y = 2(x + 5)^2 - 4\) to the general vertex form equation \(y = a(x - h)^2 + k\).
Thus, the coordinates of the vertex are \((h, k) = (-5, -4)\).
We have identified that the vertex \((h, k)\) of the parabola \(y = 2(x + 5)^2 - 4\) is \((-5, -4)\). This vertex is the lowest point on the parabola since the coefficient of the squared term is positive, indicating that the parabola opens upwards. The final coordinates are \(\boxed{(-5, -4)}\).
In real-world applications, parabolas often describe the trajectory of objects thrown into the air, such as a basketball shot or a kicked ball. The vertex of such a parabola represents the highest point that the object reaches during its flight. By identifying the vertex of the function that models a path, engineers and athletes can determine the maximum height or depth of a specific design, ensuring safety, accuracy, or optimal performance in sports and architecture.