Increasing, decreasing, maximum and minimum
A graph shows how an output (y) depends on an input (x).
- Increasing: when x gets bigger, y gets bigger (the graph goes up to the right).
- Decreasing: when x gets bigger, y gets smaller (the graph goes down).
- Constant: y stays the same (a flat part).
A maximum is a highest point; a minimum is a lowest point. Where a graph changes from rising to falling, it has a maximum.
We write intervals with brackets: [2, 6] means all x from 2 to 6, including 2 and 6. (2, 6) leaves out the two ends. In the 3D, the temperature is increasing on [0, 6] and decreasing on [6, 12].
Differences in a table
A table gives x and y in a list. To see how y changes, subtract each value from the next one. These are the differences.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 5 | 7 | 9 | 11 | 13 |
| difference | +2 | +2 | +2 | +2 |
Equal differences (for equal steps in x) mean steady growth: the graph is a straight line. If the differences keep getting bigger, the growth is speeding up (increasing growth). If they get smaller, the growth is slowing down (decreasing growth), even if y still goes up.
Increment diagrams
An increment is a small step. An increment diagram draws the steps as bars: each bar shows how much y changes in one step of x.
- Bar above the line: y went up in that step.
- Bar below the line: y went down.
- Equal bars: steady change. Growing bars: speeding up.
For one interval from a to b we write Δx = b − a (change in input) and Δy = f(b) − f(a) (change in output). The Greek letter Δ (delta) just means "change in". In the 3D, Δx is the orange line across and Δy is the purple line up.
Average rate of change (difference quotient)
The average rate of change on the interval [a, b] is
Δy ÷ Δx = (f(b) − f(a)) ÷ (b − a)
It tells you how much y changes on average for each one-unit step of x. It is also called the difference quotient.
You can find it from three sources:
- Graph: read f(a) and f(b), then divide.
- Table: take the two rows and divide the differences.
- Formula: put a and b into f(x), then divide.
The answer has a unit: (unit of y) per (unit of x), for example °C per hour or km per hour. A positive rate means y rose on average, a negative rate means it fell, and zero means it ended where it began.
Geometrically, it is the slope of the straight line (secant) through the two points A and B.
Slope of a graph at a point
On a curve the rate keeps changing, so one number for the whole graph does not exist. But at one point we can ask: how steep is the graph right here?
Take A and move B closer and closer to A. The average rate over the shorter and shorter interval settles near one number. That number is the slope of the graph at A. For f(x) = x² at x = 3 the averages over [3, 3.1] and [3, 3.01] are 6.1 and 6.01, so the slope is 6.
Where the graph is at a maximum or minimum, the slope is 0 (the graph is flat for a moment). This idea leads to the derivative in later classes.
Sequences: recursive and direct formulas
A sequence is a list of numbers in order: a₁, a₂, a₃, … Each number is a term.
- Recursive formula: gives the next term from the one before. Example: a₁ = 3, aₙ₊₁ = aₙ + 4 gives 3, 7, 11, 15, …
- Direct formula: gives any term straight from its number n. Example: aₙ = 3 + 4(n − 1), so a₁₀ = 39.
Linear (arithmetic) sequence: the differences are constant, aₙ = a₁ + d(n − 1), where d is the difference. The rate of change is d per step.
Exponential (geometric) sequence: each term is the one before multiplied by the same factor g, aₙ = a₁ × gⁿ⁻¹. The differences grow (if g > 1) or shrink in size (if 0 < g < 1). Example: 100, 200, 400, 800, … has g = 2.
Key formulas and definitions
- Δx = b − a (change in input)
- Δy = f(b) − f(a) (change in output)
- Average rate of change = Δy ÷ Δx = (f(b) − f(a)) ÷ (b − a)
- Linear sequence: aₙ = a₁ + d(n − 1); recursive: aₙ₊₁ = aₙ + d
- Exponential sequence: aₙ = a₁ × gⁿ⁻¹; recursive: aₙ₊₁ = g × aₙ
- Unit of rate = unit of y per unit of x
Worked examples
1. Easy. A plant is 5 cm tall at day 0 and 7, 9, 11, 13 cm on days 1 to 4. Are the differences equal? What is the average rate of growth?
Differences: 7 − 5 = 2, 9 − 7 = 2, 11 − 9 = 2, 13 − 11 = 2. They are equal, so the growth is steady (linear). Average rate = (13 − 5) ÷ (4 − 0) = 8 ÷ 4 = 2 cm per day.
2. Easy. The temperature graph in the 3D has f(2) = 16 °C and f(4) = 18.4 °C. Find the average rate on [2, 4].
Δx = 4 − 2 = 2 h. Δy = 18.4 − 16 = 2.4 °C. Rate = 2.4 ÷ 2 = 1.2 °C per hour (positive: it is getting warmer).
3. Medium. A bus's odometer reads 120 km at 10:00 and 295 km at 12:30. Find the average speed.
Δy = 295 − 120 = 175 km. Δx = 2.5 h. Rate = 175 ÷ 2.5 = 70 km per hour.
4. Medium. For f(x) = x², find the average rate of change on [1, 4].
f(1) = 1 and f(4) = 16. Δy = 15, Δx = 3. Rate = 15 ÷ 3 = 5.
5. Medium. Bacteria are counted as 100, 200, 400, 800. Describe the change and give the 5th term and a direct formula.
Differences: 100, 200, 400. They are growing, so growth is speeding up. Each term is 2 times the one before, so it is an exponential sequence with g = 2. The 5th term is 800 × 2 = 1600. Direct formula: aₙ = 100 × 2ⁿ⁻¹.
6. Medium. A sequence has a₁ = 3 and aₙ₊₁ = aₙ + 4. Find a₁₀.
It is linear with d = 4. Direct formula: aₙ = 3 + 4(n − 1). a₁₀ = 3 + 4 × 9 = 39.
7. Hard. For f(x) = −x² + 6x, find the average rate of change on [1, 5]. What does the answer say, and what does it not say?
f(1) = −1 + 6 = 5 and f(5) = −25 + 30 = 5. Δy = 0, so the average rate is 0 ÷ 4 = 0. It says the function ends where it started. It does not say the graph is flat: it rises to a maximum at x = 3 (f = 9) and then falls back.
8. Hard. For f(x) = x², estimate the slope at x = 3 using shorter and shorter intervals starting at 3.
[3, 3.1]: (9.61 − 9) ÷ 0.1 = 6.1. [3, 3.01]: (9.0601 − 9) ÷ 0.01 = 6.01. [3, 3.001]: about 6.001. The values settle near 6, so the slope of the graph at x = 3 is 6.
Common mistakes
- Subtracting in a different order on top and bottom. Use f(b) − f(a) and b − a (same order). If you swap both, the answer stays the same; if you swap only one, the sign is wrong.
- Forgetting the unit. The rate is "°C per hour" or "km per hour", not just a number.
- Thinking an average rate of 0 means the graph is flat. The graph may go up and then come back down.
- Calling growth "decreasing" when y is falling but the rate is just getting smaller. Check the sign of Δy, not only the size of the steps.