Identifying Increasing and Decreasing Intervals
Identifying increasing and decreasing intervals is a Grade 11 functions skill in Big Ideas Math. A function is increasing on an interval when, as x increases, f(x) also increases (graph rises left to right). It is decreasing when f(x) decreases as x increases (graph falls). To identify these intervals, analyze the graph visually or find where the derivative is positive (increasing) or negative (decreasing). Intervals are expressed in interval notation: e.g., increasing on (−2, 1) and decreasing on (1, 4). For polynomial functions, critical points (where the derivative equals zero) mark the boundaries between increasing and decreasing intervals.
Key Concepts
A function is increasing on an interval if as $x$ values move from left to right, the $y$ values rise (positive slope). A function is decreasing on an interval if as $x$ values move from left to right, the $y$ values fall (negative slope).
Common Questions
What does it mean for a function to be increasing on an interval?
A function is increasing on an interval if, for any two x-values where x₁ < x₂, we have f(x₁) < f(x₂). The graph rises from left to right over that interval.
What does it mean for a function to be decreasing on an interval?
A function is decreasing on an interval if, for any x₁ < x₂, we have f(x₁) > f(x₂). The graph falls from left to right over that interval.
How do you identify increasing and decreasing intervals from a graph?
Read the graph from left to right. Mark where the graph starts rising (beginning of increasing interval) and where it peaks or begins falling (end of increasing interval, start of decreasing).
How are increasing and decreasing intervals written in interval notation?
Use parentheses (open intervals) at the boundaries since functions are neither increasing nor decreasing at isolated points: e.g., increasing on (−2, 1), decreasing on (1, 4).
What are critical points and how do they relate to increasing/decreasing intervals?
Critical points occur where the rate of change equals zero or is undefined. They are the x-values that mark boundaries between increasing and decreasing behavior.
Can a function be both increasing and decreasing at the same x-value?
No—at any specific point, a function is classified based on the interval behavior around it. A local maximum is where a function transitions from increasing to decreasing.