How to Find Increasing and Decreasing Intervals on Graphing Calculator

Finding increasing and decreasing intervals of a function is a fundamental calculus skill. These intervals tell you where a function is rising or falling, which is crucial for understanding the behavior of functions in calculus and real-world applications. This guide explains how to find these intervals using a graphing calculator.

What Are Increasing and Decreasing Intervals?

In calculus, a function's increasing and decreasing intervals describe where the function is rising or falling. These intervals are determined by the first derivative of the function:

If f'(x) > 0 on an interval, the function is increasing on that interval.
If f'(x) < 0 on an interval, the function is decreasing on that interval.
If f'(x) = 0 on an interval, the function is constant on that interval.

Critical points (where f'(x) = 0 or is undefined) divide the domain into intervals where the function's behavior can be analyzed.

How to Find Increasing and Decreasing Intervals

Step 1: Find the First Derivative

Start by finding the first derivative of the function. This will help you determine where the function is increasing or decreasing.

Step 2: Find Critical Points

Set the first derivative equal to zero or undefined to find critical points. These points divide the domain into intervals.

Step 3: Test Intervals

Choose test points in each interval and evaluate the sign of the first derivative. This tells you whether the function is increasing or decreasing on that interval.

Step 4: Analyze Results

Based on the test results, determine the increasing and decreasing intervals. Document your findings clearly.

Using a Graphing Calculator

A graphing calculator can help visualize and find increasing and decreasing intervals more efficiently. Here's how to use one:

Step 1: Graph the Function

Enter the function into the graphing calculator and graph it. This helps you visualize the function's behavior.

Step 2: Find the First Derivative

Use the calculator to find the first derivative of the function. This can often be done using the derivative function or by manually entering the derivative.

Step 3: Find Critical Points

Set the derivative equal to zero or undefined and solve for x. These are the critical points that divide the intervals.

Step 4: Test Intervals

Choose test points in each interval and evaluate the sign of the derivative. This tells you whether the function is increasing or decreasing on that interval.

Step 5: Document Results

Clearly document the increasing and decreasing intervals based on the test results.

Tip: Many graphing calculators have built-in features to find critical points and test intervals automatically. Check your calculator's manual for specific instructions.

Worked Example

Let's find the increasing and decreasing intervals for the function f(x) = x³ - 3x².

Step 1: Find the First Derivative

f'(x) = d/dx (x³ - 3x²) = 3x² - 6x

Step 2: Find Critical Points

Set f'(x) = 0:

3x² - 6x = 0
3x(x - 2) = 0
x = 0 or x = 2

Step 3: Test Intervals

Test points in each interval:

For x < 0 (e.g., x = -1): f'(-1) = 3(-1)² - 6(-1) = 3 + 6 = 9 > 0 → Increasing
For 0 < x < 2 (e.g., x = 1): f'(1) = 3(1)² - 6(1) = 3 - 6 = -3 < 0 → Decreasing
For x > 2 (e.g., x = 3): f'(3) = 3(3)² - 6(3) = 27 - 18 = 9 > 0 → Increasing

Step 4: Document Results

The function is increasing on (-∞, 0) and (2, ∞), and decreasing on (0, 2).

FAQ

What if the derivative is zero over an entire interval?: The function is constant on that interval, meaning it's neither increasing nor decreasing.
How do I know if a critical point is increasing or decreasing?: Critical points are where the derivative changes sign. If the derivative changes from positive to negative, the function has a local maximum. If it changes from negative to positive, the function has a local minimum.
Can I use a graphing calculator to find increasing and decreasing intervals automatically?: Many graphing calculators have built-in features to find critical points and test intervals automatically. Check your calculator's manual for specific instructions.
What if the function has a vertical asymptote?: The derivative will be undefined at the vertical asymptote. You can still find increasing and decreasing intervals by testing points on either side of the asymptote.