Lesson 3 ~45 min Rates of Change · Path +90 XP

Variable Rates and Curves

A curved graph is describing a rate that will not stay still. There are exactly four ways that can happen, each with its own shape and its own standard phrase, and learning to say which one you are looking at is the whole of this lesson.

Today's hook: "Increasing at a decreasing rate" sounds like a contradiction and is not. Something is going up, and it is going up more slowly than it was. A cooling cup of tea, a filling bathtub with a narrowing top, a savings balance whose interest is falling: the phrase names a shape you have seen many times.

0/5QUESTS

Think First

warm-up

A tree grows quickly when young and more slowly as it matures, but it never shrinks. Sketch its height against time. Now describe your sketch in two words: is the height increasing or decreasing, and is the rate of growth increasing or decreasing? Those two answers together name the shape.

Record your answer in your workbook.

The Big Idea

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A curve is describing a variable rate, and there are exactly four possibilities. The description always has two parts: the first word says whether the quantity is rising or falling, and the second says whether the steepness is growing or shrinking.

$$\text{increasing or decreasing} \ \times \ \text{at an increasing or decreasing rate}$$

The two parts are independent, which is why there are four combinations and not two. A quantity can fall quickly and then more slowly, or slowly and then more quickly, and those are different situations that need different words.

$\text{direction} \times \text{steepness}$

Answer two questions

Is it going up or down? Is it getting steeper or flatter? Then combine.

Steepness, not height

The second part is about how the slope changes, not about whether the graph is high or low.

Compare equal intervals

Look at successive equal steps across. Are the steps up getting bigger or smaller?

What You'll Master

objectives

Know

That a curved graph represents a rate of change that is not constant
The four qualitative descriptions and the shape each corresponds to
That the direction and the change in steepness are independent

Understand

Why "increasing at a decreasing rate" is not a contradiction
Why a curved section has no single rate but does have an average

Can Do

Describe a curved graph using the standard two-part phrase
Match a described situation to one of the four shapes
Compare rates at different points on a curve by comparing steepness

Words You Need

vocabulary

Variable rateA rate of change that differs from one moment to another.

Qualitative descriptionA description in words rather than numbers.

Increasing rateThe graph is getting steeper, whichever direction it runs in.

Decreasing rateThe graph is getting flatter, whichever direction it runs in.

Levelling offIncreasing at a decreasing rate, so the graph approaches a horizontal direction.

Two Questions, Four Answers

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Describing a curved graph means answering two separate questions.

Question 1: which way is it going? If the graph rises from left to right, the quantity is increasing. If it falls, the quantity is decreasing.

Question 2: what is the steepness doing? If the curve is getting steeper, the rate is increasing. If it is getting flatter, the rate is decreasing.

The answers are independent, so there are four combinations, which are the four panels in the diagram above:

increasing at an increasing rate;
increasing at a decreasing rate;
decreasing at an increasing rate;
decreasing at a decreasing rate.

The second question is about the slope, not the height. A graph can be very high up and still be flattening out, and it can be near the bottom of the page and still be getting steeper. Confusing the two is the commonest error here, and asking the two questions separately is what prevents it.

Reading Steepness From Equal Intervals

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Deciding whether a curve is steepening or flattening is easier with a systematic method than by eye.

Take several equal steps along the horizontal axis and look at the vertical change over each.

If the vertical changes are getting larger, the curve is getting steeper and the rate is increasing.
If they are getting smaller, the curve is flattening and the rate is decreasing.

For a graph with values $0, 1, 4, 9, 16$ at times $0, 1, 2, 3, 4$: the changes are $1, 3, 5, 7$. They are growing, so this is increasing at an increasing rate.

For values $0, 10, 15, 17.5, 18.75$: the changes are $10, 5, 2.5, 1.25$. They are shrinking, so this is increasing at a decreasing rate.

The same method works on a falling graph, using the sizes of the drops. For values $100, 50, 25, 12.5$: the drops are $50, 25, 12.5$, which are shrinking, so this is decreasing at a decreasing rate.

Note what that last example is not. The quantity is falling fast at first and then more gently, which is a decreasing rate even though the numbers themselves keep going down. Direction and steepness are answering different questions.

What the Four Shapes Look Like in Life

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Increasing at an increasing rate. Something growing ever faster: a population with nothing limiting it, a compound-interest balance, the distance covered by something accelerating. The curve sweeps upward.

Increasing at a decreasing rate. Something growing but levelling off: a tree's height, a bath filling through a tap that is slowly closing, the total distance of a vehicle that is slowing but still moving forward. This is the shape most often described as "levelling off" or "approaching a limit".

Decreasing at an increasing rate. Something falling ever faster: an object gathering speed as it falls, a balance being drained by growing withdrawals. The curve plunges.

Decreasing at a decreasing rate. Something falling but flattening out: a hot drink cooling towards room temperature, a bouncing ball's height between bounces, a car braking gently to a stop. It approaches a value without dropping sharply at the end.

The two "decreasing rate" shapes both level off, and the two "increasing rate" shapes both run away. The second word predicts the long-run behaviour, which is often what a question really wants.

Cooling is the standard example

A hot drink cools quickly at first, when it is far above room temperature, and more slowly as it gets closer. So its temperature is decreasing at a decreasing rate, and it flattens towards room temperature without ever quite reaching it. That last detail makes it a good example of an asymptote in a real setting.

What You Can and Cannot Quote

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A curved section has no single rate, so there is no number that is "the rate". Three honest things can still be said.

The qualitative description. One of the four phrases. This is what most questions ask for, and it is a complete answer.

The average rate over an interval. Take the two endpoints of the interval and compute the gradient of the straight line joining them. This is a genuine number and should be labelled as an average.

$$\text{average rate} = \frac{\text{change in the quantity}}{\text{length of the interval}}$$

A comparison between two moments. The curve is steeper at one point than another, so the rate is larger there. This needs no calculation at all and is often the fastest way to answer.

What cannot be done at this level is finding the exact rate at a single instant, since that would need the gradient at a point rather than over an interval. Attempting it by taking two very close points is the right instinct, and it is exactly the idea that calculus makes precise in Year 11.

So a complete answer to "describe the rate of change" on a curve is a phrase, not a number, unless an interval is specified.

Common Pitfalls

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Reading "increasing at a decreasing rate" as a contradiction, or as meaning the quantity decreases.

Fix: the first word is the direction and the second is the steepness. The quantity goes up; it just goes up more gently as it goes.

Judging the second part from how high the graph is rather than from its slope.

Fix: compare the vertical change over successive equal horizontal steps. Growing changes mean an increasing rate, whatever height the graph is at.

Quoting a single rate for a curved section.

Fix: describe it in words, or give the average over a stated interval and call it an average. There is no single value to give.

Describing a falling curve that flattens as "decreasing at an increasing rate" because the values keep dropping.

Fix: the drops are getting smaller, so the rate is decreasing. The values continuing to fall is the first word, not the second.

Watch Me Solve It · 3 examples

Watch Me Solve It · Describing from a table

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PROBLEM

A quantity takes the values $200, 140, 104, 82.4$ at times $0, 1, 2, 3$. Describe how it is changing, using the standard two-part phrase, and justify each part.

1

Answer the direction question

The values fall throughout, from $200$ down to $82.4$, so the quantity is decreasing.
2

Find the change over each equal interval

$200 \to 140: \ -60$

$140 \to 104: \ -36$

$104 \to 82.4: \ -21.6$

The time steps are all one unit, so these are comparable.
3

Answer the steepness question

The sizes of the drops are $60$, $36$ and $21.6$, which are getting smaller. So the graph is flattening and the rate is decreasing.
4

Combine and check against a context

The quantity is decreasing at a decreasing rate. This is the cooling shape: it falls quickly at first and levels off, approaching some value from above. Here each drop is $0.6$ of the previous one, so the values approach a limit rather than continuing to zero at a steady pace.

AnswerDecreasing at a decreasing rate

Watch Me Solve It · Matching situations to shapes

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PROBLEM

Describe the shape of the graph for each situation: (a) the height of a sunflower over a season; (b) the speed of a stone dropped from a cliff; (c) the amount of medicine remaining in the body after a dose; (d) the number of people who have heard a rumour, early on.

1

(a) Sunflower height

It grows throughout, so increasing. It grows fast when young and slows as it matures, so the rate is decreasing. Increasing at a decreasing rate: the graph levels off.
2

(b) Speed of a falling stone

The speed rises throughout, so increasing. Ignoring air resistance it gains speed at a steady rate, which would be a straight line rather than a curve; with air resistance the gain slows as it falls faster, giving increasing at a decreasing rate. Naming the assumption is part of the answer.
3

(c) Medicine remaining

It falls throughout, so decreasing. The body clears a proportion of what is present, so more is cleared per hour when there is more present, and the drops shrink over time. Decreasing at a decreasing rate.
4

(d) A rumour spreading, early on

The number rises, so increasing. Early on, each person who knows can tell others, so the more who know the faster it spreads. Increasing at an increasing rate. Later it would level off as almost everyone has heard, changing to increasing at a decreasing rate, so the description depends on which phase is being described.

Answer(a) and (b) increasing at a decreasing rate; (c) decreasing at a decreasing rate; (d) increasing at an increasing rate, early on

Watch Me Solve It · Average rate and comparison

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PROBLEM

A curve of volume in litres against time in minutes passes through $(0, 0)$, $(2, 30)$, $(4, 45)$ and $(6, 52)$. (a) Describe the shape. (b) Find the average rate over the first two minutes and over the last two. (c) Explain how (b) confirms (a). (d) State whether the rate at $t = 6$ is above or below the overall average.

1

(a) Answer both questions

The volume rises throughout, so increasing. The rises over successive two-minute intervals are $30$, $15$ and $7$, which are shrinking, so the rate is decreasing. Increasing at a decreasing rate.
2

(b) Average over each interval

$\frac{30-0}{2} = 15, \qquad \frac{52-45}{2} = 3.5$

So $15$ litres per minute over the first two minutes, and $3.5$ litres per minute over the last two.
3

(c) Connect the numbers to the shape

The average rate has fallen from $15$ to $3.5$ litres per minute, which is exactly what "at a decreasing rate" claims. The description and the calculation agree.
4

(d) Compare with the overall average

$\frac{52 - 0}{6} \approx 8.67 \ \text{litres per minute}$

The rate at $t = 6$ is below that. Since the rate falls throughout, it is above average early and below average late; and the interval average of $3.5$ near the end is already below $8.67$, so the instantaneous rate there is lower still.

Answer(a) increasing at a decreasing rate; (b) $15$ then $3.5$ L/min; (d) below the overall average of about $8.67$ L/min

Brain Trainer · 5 problems

Brain Trainer · Name the shape

5 problems

Five items on qualitative descriptions. Work each one, then reveal the answer.

1 A graph rises and gets steeper. Describe it.

Direction from the rise, steepness from the curvature.Increasing at an increasing rate
2 A graph rises and flattens out. Describe it.

Still going up, but more gently.Increasing at a decreasing rate
3 Values are $80, 40, 20, 10$ at equal time steps. Describe the change.

Drops of $40$, $20$, $10$: shrinking.Decreasing at a decreasing rate
4 Values are $2, 4, 8, 16$ at equal time steps. Describe the change.

Rises of $2$, $4$, $8$: growing.Increasing at an increasing rate
5 Can you quote a single rate for a curved section?

The gradient differs at every point.No, only an average over an interval

Complete in your workbook.

Multiple Choice · 5 questions

MC1

The phrase

+10 XP

"Increasing at a decreasing rate" describes a graph that:

MC2

From a table

+10 XP

A quantity takes the values $5, 9, 17, 33$ at equally spaced times. It is:

MC3

Falling and flattening

+10 XP

A hot drink cools towards room temperature. Its temperature is:

MC4

Steepness against height

+10 XP

A graph is very high on the page and is flattening out. Its rate of change is:

MC5

What can be quoted

+10 XP

For a curved section of a graph, you may correctly state:

Short Answer · 3 questions

Describe from data

+15 XP

SHORT ANSWER

For each set of values, taken at equally spaced times, state the two-part description and justify it with the differences.
(a) $3, 7, 11, 15$
(b) $100, 60, 36, 21.6$
(c) $1, 3, 9, 27$
(d) $50, 74, 86, 92$

Write your working in your book.

Situations and shapes

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SHORT ANSWER

For each situation, state which of the four descriptions applies and sketch in words what the graph looks like.
(a) The depth of water in a bath being filled by a tap running at a steady rate, where the bath is wider at the top.
(b) The value of a car over the years after purchase.
(c) The total distance covered by a train pulling out of a station.
(d) The number of bacteria in a dish with unlimited food, in the early stages.

Write your working in your book.

Numbers on a curve

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SHORT ANSWER

A curve of temperature in degrees Celsius against time in minutes passes through $(0, 90)$, $(5, 60)$, $(10, 42)$, $(15, 32)$ and $(20, 26)$.
(a) Describe the change using the standard phrase, justifying it.
(b) Find the average rate of change over the first five minutes and over the last five.
(c) Explain what those two numbers show about the shape.
(d) Estimate the room temperature the drink is approaching, and explain how you can tell it will never quite reach it.

Write your working in your book.

Stretch Challenge · Where the shapes come from

+25 XP

CHALLENGE

(a) Explain why a quantity that grows by a fixed proportion each time period is increasing at an increasing rate, while one that grows by a fixed amount is increasing at a constant rate.
(b) A quantity decreases at a decreasing rate and approaches a limit $L$. Explain why the graph of the difference between the quantity and $L$ is decreasing at a decreasing rate too, and why it approaches zero.
(c) Can a graph be increasing at an increasing rate over one interval and increasing at a decreasing rate over another? If so, describe such a graph and name the point where the change happens.

(a) Growing by a fixed amount means each period adds the same number, so the differences are all equal, the graph is a straight line, and the rate is constant. Growing by a fixed proportion means each period adds a percentage of the current value: the amount added is proportional to what is already there. Since the quantity is growing, the base it is calculated from is growing too, so the amount added grows each period. Growing increments mean a steepening graph, so the quantity is increasing at an increasing rate. This is the difference between simple and compound interest, and it is why compound growth eventually outruns any fixed-amount growth however large that fixed amount is.

(b) Let $g$ be the gap, $g = Q - L$, where $Q$ is the quantity. Since $L$ is a constant, subtracting it shifts the whole graph down by $L$ without changing its shape at all: every difference between successive values of $g$ equals the corresponding difference for $Q$. So the gap falls, and the sizes of its drops shrink exactly as $Q$'s do, meaning $g$ is also decreasing at a decreasing rate. It approaches zero because $Q$ approaches $L$, and $g = Q - L$ approaches $L - L = 0$. This is a genuinely useful move in practice: plotting the gap rather than the raw quantity often reveals a simple pattern, since the gap frequently shrinks by a constant proportion each period even when the raw values do not.

(c) Yes. Such a graph rises throughout, steepening at first and then flattening, so it is shaped like a stretched letter S. The point where the behaviour changes is called a point of inflection: it is where the curve stops bending one way and starts bending the other, and it is where the graph is at its steepest. Real examples are common: a rumour or an epidemic spreads slowly at first, then explosively as many people are involved, then slowly again as almost everyone has been reached; the total sales of a new product often follow the same pattern. Describing such a graph honestly requires naming both phases and where they meet, rather than forcing a single phrase onto the whole curve. The same term, point of inflection, appeared in the Polynomials focus area at a triple zero, and it means the same thing here.

Quick Review

recap

Two questions

Direction from the rise or fall; rate from the steepness

Four combinations

Both parts vary independently

Equal intervals

Compare successive changes to judge the steepness

No single rate

Describe in words, or average over a stated interval

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