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What is the Decreasing Interval of a Function?
Grade Level:
Class 10
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The decreasing interval of a function is a range of 'x' values where the function's output (its 'y' value) continuously goes down as you move from left to right on its graph. Imagine walking along the graph; if you're going downhill, you're in a decreasing interval.
Simple Example
Quick Example
Think about the price of mobile data per GB in India over the last few years. If we plot 'Year' on the x-axis and 'Price' on the y-axis, the graph would generally show prices falling as years pass. This period where prices are continuously dropping represents a decreasing interval of the 'price over time' function.
Worked Example
Step-by-Step
Let's consider a function that describes the temperature in a city over 24 hours, represented by a simplified graph. We want to find the decreasing interval.
Step 1: Look at the graph of the function from left to right (as 'x' values increase).
---Step 2: Identify the portion of the graph where the line is sloping downwards.
---Step 3: Suppose the temperature starts at 25 degrees at x=0 (midnight), rises to 35 degrees at x=6 (6 AM), then falls to 20 degrees at x=18 (6 PM), and rises back to 25 degrees at x=24 (midnight).
---Step 4: From x=0 to x=6, the temperature is rising. This is an increasing interval.
---Step 5: From x=6 to x=18, the temperature is falling from 35 degrees to 20 degrees. This is a decreasing interval.
---Step 6: From x=18 to x=24, the temperature is rising again. This is an increasing interval.
---Step 7: The decreasing interval for this temperature function is from x=6 to x=18.
Answer: The decreasing interval is (6, 18).
Why It Matters
Understanding decreasing intervals helps engineers design systems that cool down efficiently, or economists predict when market prices might fall. Data scientists use this to find trends in data, like when the usage of a particular app is declining, which is crucial for making business decisions.
Common Mistakes
MISTAKE: Confusing decreasing intervals with negative function values. A function can be decreasing even if its y-values are positive. | CORRECTION: A decreasing interval means the graph is going down, not that the graph is below the x-axis.
MISTAKE: Including endpoints in the interval when the function is flat or turning. | CORRECTION: At the exact point where a function changes from increasing to decreasing (or vice versa), it is neither increasing nor decreasing. So, intervals are usually written with open brackets, like (a, b).
MISTAKE: Reading the interval from top to bottom on the y-axis instead of left to right on the x-axis. | CORRECTION: Always read the x-axis values from left to right to determine the interval.
Practice Questions
Try It Yourself
QUESTION: For a function f(x) where the graph goes up from x=0 to x=5, then goes down from x=5 to x=10, and then goes up again from x=10 to x=15, what is the decreasing interval? | ANSWER: (5, 10)
QUESTION: A function's value is 10 at x=2, 8 at x=4, 5 at x=6, and 7 at x=8. Assuming the trend is continuous between these points, in which interval is the function decreasing? | ANSWER: (2, 6)
QUESTION: The speed of a delivery scooter (in km/h) is given by S(t) = 0.5t^2 - 10t + 50, where 't' is time in minutes (from t=0 to t=20). Find the interval where the scooter's speed is decreasing. (Hint: The lowest point of a parabola y = ax^2 + bx + c is at x = -b/(2a)). | ANSWER: (0, 10)
MCQ
Quick Quiz
Which of the following describes a decreasing interval of a function?
The function's y-values are negative.
As x increases, the y-values decrease.
The graph of the function is flat.
As x decreases, the y-values decrease.
The Correct Answer Is:
B
A decreasing interval means that as you move along the x-axis to the right (x increases), the function's value (y) goes down. Option B correctly describes this. Options A, C, and D describe different characteristics.
Real World Connection
In the Real World
In cricket analytics, if you plot a batsman's strike rate (y-axis) against the overs played (x-axis) in a match, a decreasing interval would show a phase where the batsman's scoring speed is going down. Coaches and analysts use this to understand performance trends and make strategic decisions during a game.
Key Vocabulary
Key Terms
INTERVAL: A range of numbers between two given numbers, often represented with brackets | FUNCTION: A rule that assigns exactly one output to each input | GRAPH: A visual representation of a function, usually on a coordinate plane | X-AXIS: The horizontal number line on a graph | Y-AXIS: The vertical number line on a graph
What's Next
What to Learn Next
Great job understanding decreasing intervals! Next, you should learn about 'Increasing Intervals of a Function' and 'Constant Intervals'. Knowing all three will help you fully describe the behavior of any function's graph.
