Practice Reading Monotonicity from a Graph
Step-by-step explanation, worked examples, and unlimited practice.
📖 Pre-Calculus – Monotonicity | Grade 11 Math
Pre-Calculus: Reading Graphs
Monotonicity – Increasing and Decreasing
🎯 What Is Monotonicity?
Monotonicity describes the behaviour of the function: is it increasing or decreasing?
As we move along the x-axis from left to right, what happens to the y-values?
📈 Increasing Function
As x increases, y also increases
Move right ↗ go up
💡 Imagine: climbing a hill — the further you go, the higher you get!
📝 Mathematical definition:
If \(x_1 < x_2\) then \(f(x_1) < f(x_2)\)
📉 Decreasing Function
As x increases, y decreases
Move right ↘ go down
💡 Imagine: going down a slide — the further you go, the lower you get!
📝 Mathematical definition:
If \(x_1 < x_2\) then \(f(x_1) > f(x_2)\)
📊 Intervals of Increase and Decrease
Most functions do not increase or decrease all the time — they have intervals of increase and decrease.
✏️ In this graph:
Decreasing: on interval \((-\infty, -2)\)
Increasing: on interval \((-2, 1)\)
Decreasing: on interval \((1, \infty)\)
⚠️ Important!
Intervals are written in x-values (not y-values!)
🔍 How to Identify Monotonicity from a Graph?
💡 Trick: imagine walking along the graph from left to right
Going up? 📈
Like climbing a hill
= increasing function
Going down? 📉
Like descending a slope
= decreasing function
➡️ Constant Function
There is a third case: the function neither increases nor decreases — it is constant.
The y-value stays the same for all x
✏️ Full Example
Graph of \(f(x) = x^2\)
Monotonicity intervals:
Decreasing: on interval \((-\infty, 0)\)
Increasing: on interval \((0, \infty)\)
Transition point: x = 0 (minimum point)
📝 Summary
Increasing 📈 = move right, go up
Decreasing 📉 = move right, go down
Monotonicity intervals written in x-values
Transition points = extrema (in the next page!)
Worked Examples
📊 Exercise:
A graph is increasing on the interval \((-\infty, 2)\) and decreasing on the interval \((2, \infty)\). What happens at \(x=2\)?
Show solution
| Monotonicity analysis The situation: Increasing: \((-\infty, 2)\) Decreasing: \((2, \infty)\) What happens at \(x=2\)? Analysis: 1️⃣ Before \(x=2\): increasing ↗ 2️⃣ At \(x=2\): transition point 3️⃣ After \(x=2\): decreasing ↘ Conclusion: Increasing → Decreasing This is a local maximum point! 🔝 Reasoning: The function increased up to \(x=2\) Reached a peak Then decreased → \(x=2\) is the highest point! |
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