- Identify symmetry in the provided data points to locate the vertex at \(x = 1\).
- Determine that the function opens downward since the vertex is a maximum at \(k(1) = 9\).
- Conclude that a downward-opening parabola is increasing on the interval to the left of the vertex.
- The correct interval is \(\boxed{(-\infty, 1)}\).
Explanation
- Analysis of the Function Data
The problem asks to identify the interval over which the given quadratic function \(k(x)\) is increasing. We are provided with a set of points: \((-1, 5)\), \((0, 8)\), \((1, 9)\), \((2, 8)\), \((3, 5)\), and \((4, 0)\). Since \(k(x)\) is a quadratic function, it follows the form \(k(x) = ax^2 + bx + c\), and its graph is a parabola.
- Finding the Axis of Symmetry and Vertex
To determine the properties of the parabola, we look for its axis of symmetry. Observing the points, we see that the function values are symmetric around \(x = 1\). Specifically, \(k(0) = k(2) = 8\) and \(k(-1) = k(3) = 5\). This symmetry implies that the vertex of the parabola is located at the x-coordinate \(h = 1\). The value at this vertex is \(k(1) = 9\), which is the maximum point given the symmetry of the surrounding values. Since the function has a maximum, the parabola opens downward, meaning the leading coefficient \(a\) is negative.
- Conclusion and Interval Selection
A parabola that opens downward increases as we move from left to right until it reaches its vertex and then decreases. Since our vertex is at \(x = 1\), the function increases for all values of \(x\) less than 1. Therefore, the interval of the domain over which the function is increasing is \((-\infty, 1)\). Comparing this to the given options, we select the first one.
Examples
Understanding where a function is increasing or decreasing is crucial in fields like economics for modeling profit or cost functions. For example, if a company's profit is modeled by a quadratic function, identifying the interval of growth helps business owners understand the range of production levels over which their profit is increasing, allowing them to optimize their output. This visual and algebraic analysis allows for better decision-making in real-world scenarios.