Question

the function f is graphed below. determine the intervals on which f increasing and decreasing.

Explanation:

Step1: Recall Increasing/Decreasing Definition

A function \( f(x) \) is increasing on an interval if, as \( x \) increases, \( f(x) \) increases (graph rises from left to right). It is decreasing if, as \( x \) increases, \( f(x) \) decreases (graph falls from left to right).

Step2: Analyze Left Segment (x < -2)

From the leftmost point (around \( x = -8 \)) to \( x = -2 \), the graph falls as \( x \) increases. So \( f \) is decreasing on \( (-\infty, -2) \).

Step3: Analyze Middle Segment (-2 < x < 4)

From \( x = -2 \) to \( x = 4 \), the graph rises as \( x \) increases (from the vertex at \( x = -2 \) up to the vertex at \( x = 4 \)). So \( f \) is increasing on \( (-2, 4) \).

Step4: Analyze Right Segment (x > 4)

From \( x = 4 \) to the rightmost point (around \( x = 7 \)), the graph falls as \( x \) increases. So \( f \) is decreasing on \( (4, \infty) \).

Answer:

Increasing interval: \( (-2, 4) \)
Decreasing intervals: \( (-\infty, -2) \) and \( (4, \infty) \)