Question

determine the domain on which the following function is increasing.

Explanation:

Step1: Recall the definition of an increasing function

A function \(y = f(x)\) is increasing on an interval if for any two points \(x_1\) and \(x_2\) in that interval with \(x_1eq0\)), if \(a<0\) (opens down - like the given graph), it is increasing on the interval \((-\infty,-\frac{b}{2a})\) (from the left up to the vertex). Visually, we look at the part of the graph where as \(x\) increases, \(y\) also increases.

Step2: Locate the vertex \(x\) - coordinate

From the graph, we can observe the vertex (the highest point of the parabola). By counting the grid - squares, the \(x\) - coordinate of the vertex is \(x=-1\).

Step3: Determine the increasing interval

Since the parabola opens down (the coefficient of \(x^{2}\) is negative), the function is increasing for \(x\) values less than the \(x\) - coordinate of the vertex.

Answer:

The function is increasing on the interval \((-\infty,-1)\)