Question

state where the graph of f is increasing and where it is decreasing. f(x)=x² - 2x - 2 for what values of x is the graph of f increasing? x (type an inequality symbol, then type an integer.)

Explanation:

Step1: Identify the function type

The function \(f(x)=x^{2}-2x - 2\) is a quadratic function in the form \(f(x)=ax^{2}+bx + c\) where \(a = 1\), \(b=-2\), \(c=-2\). Since \(a>0\), the parabola opens upwards.

Step2: Find the vertex's x - coordinate

The x - coordinate of the vertex of a quadratic function \(f(x)=ax^{2}+bx + c\) is given by the formula \(x=-\frac{b}{2a}\). Substituting \(a = 1\) and \(b=-2\) into the formula, we get \(x=-\frac{-2}{2\times1}=\frac{2}{2}=1\).

Step3: Determine the increasing interval

For a parabola that opens upwards (\(a > 0\)), the function is increasing to the right of the vertex. So the function \(f(x)\) is increasing for \(x>1\).

Answer:

\(x > 1\)