Question

determine the domain on which the following function is increasing.

Explanation:

Step1: Recall the property of increasing function

A function \(y = f(x)\) is increasing on an interval if for any two points \(x_1\) and \(x_2\) in the interval with \(x_1eq0\)), if \(a<0\), the parabola opens down - ward.

Step2: Analyze the given parabola

The given parabola opens down - ward (since it has a "cap" at the top). The vertex of a parabola \(y = ax^{2}+bx + c\) has \(x\) - coordinate \(x =-\frac{b}{2a}\). For a parabola, the function is increasing to the left of the vertex.
From the graph, we can observe that the vertex of the parabola has \(x\) - coordinate \(x=-2\).
So the function is increasing for \(x < - 2\), which in interval notation is \((-\infty,-2)\)

Answer:

$$(-\infty, - 2)$$