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 > 1 (type an inequality symbol, then type an integer.) for what values of x is the graph of f decreasing? 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 \( a=1>0 \), 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 increasing and decreasing intervals

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

Answer:

For the decreasing part: \( x < 1 \)