# Increasing and Decreasing Intervals – Formula, Definition With Examples

Welcome to Brighterly, where we make math easy and fun for kids. Today, we’re looking at increasing and decreasing intervals. These are key to understanding graphs in math. At Brighterly, we keep learning interactive and engaging. Let’s get started and see how these intervals work in math!

## What Are Increasing and Decreasing Intervals?

In mathematics, increasing and decreasing intervals are parts of a graph where values either go up or down. When the x-values (horizontal axis) get bigger and the y-values (vertical axis) also get bigger, the graph is going up. This is an increasing interval. When the y-values get smaller as the x-values get bigger, the graph is going down. This is a decreasing interval.

## Definition of Increasing Intervals

An increasing interval is where, as you move right on the graph (increase x-values), the graph goes up (y-values increase). It’s like climbing a hill: the further you walk (x), the higher you go (y). A simple formula for this is:

If `x₂ > x₁, then y₂ > y₁`

.

Think about a business. If its sales (y) increase each year (x), the sales graph shows an increasing interval.

## Definition of Decreasing Intervals

A decreasing interval is the opposite. As x-values increase, y-values decrease. It’s like walking down a hill: as you go further (x), you go lower (y). The formula is:

If x₂ > x₁, then y₂ < y₁.

Imagine a car losing speed (y) as it runs out of fuel over time (x). This shows a decreasing interval in a speed-time graph.

## Properties of Increasing and Decreasing Intervals

### Properties of Increasing Intervals

**Continuity**: The graph doesn’t jump or break.**Positive Slope**: The graph slopes upward.**No Peaks**: You don’t find top points in this part of the graph.

### Properties of Decreasing Intervals

**Continuity**: The graph is unbroken.**Negative Slope**: The graph slopes downward.**No Valleys**: There are no low points in this section.

## Difference Between Increasing and Decreasing Intervals

The main difference is direction. In increasing intervals, graphs go up. In decreasing intervals, they go down. It’s like comparing going up an escalator versus going down.

## Practice Problems on Increasing and Decreasing Intervals

- Look at the graph of f(x) = x² – 4x + 3. Where is it increasing and decreasing?
- For f(x) = 2/x, identify the increasing and decreasing parts.

## Conclusion

Understanding increasing and decreasing intervals helps us read and interpret graphs. This skill is useful in many real-life scenarios, like analyzing business trends or understanding speed changes in vehicles.

## Frequently Asked Questions on Increasing and Decreasing Intervals

### How do you know if an interval is increasing or decreasing?

Look at the graph: if it goes up as you move right, it’s increasing. If it goes down, it’s decreasing.

### Can a graph have both types of intervals?

Yes, many graphs show both increasing and decreasing parts.

### Does the slope of the graph matter?

Yes, a positive slope means increasing, and a negative slope means decreasing.

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