Question

determine the domain on which the following function is increasing.
answer
all real numbers

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 \(x_1eq0\)), its axis of symmetry is \(x =-\frac{b}{2a}\). In the case of a parabola opening downwards (\(a<0\)), the function is increasing to the left of the axis of symmetry.

Step2: Analyze the given parabola

The given parabola is a quadratic function (a parabola). Since it opens downwards (the coefficient of \(x^{2}\) is negative). The axis of symmetry of a parabola \(y = ax^{2}+bx + c\) is \(x =-\frac{b}{2a}\). For a parabola symmetric about the \(y\) - axis (\(b = 0\)), the axis of symmetry is \(x = 0\).

Step3: Determine the increasing interval

Looking at the graph, as \(x\) moves from \(-\infty\) to \(0\), the \(y\) - values of the function are getting larger. So the function is increasing on the interval \((-\infty,0)\).

Answer:

$(-\infty,0)$