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D1.3 · Sketch graphs from verbal rate descriptions

Learn to sketch graphs from verbal rate descriptions through clear examples and targeted practice.

Ontario Grade 12 Mathematics

Characteristics of Functions

Translate descriptions of change into graph shape

A verbal description can tell you how a quantity changes without giving you a formula. To sketch its graph, identify the quantities on the axes, then decide whether the graph rises or falls and how its steepness changes. A rate compares the change in one quantity with the change in another. This lets you sketch the described behaviour even when exact values are not given.

What you will learn

Identify the quantities shown on the horizontal and vertical axes.
Use a rate’s sign to decide whether a graph rises, falls, or stays level.
Use the size of a rate and how it changes to describe steepness.
Sketch a graph that matches a verbal description of changing rates.

1. Review: axes, change, and rate

A graph shows how one quantity depends on another. The independent variable is the input quantity, shown on the horizontal axis. Time is a common example. The dependent variable is the quantity whose value changes, shown on the vertical axis.

Read a graph from left to right. It increases when it moves upward and decreases when it moves downward. If its height stays the same over an interval, the graph is horizontal there.

The rate of change compares the change in the vertical quantity with the change in the horizontal quantity. For a graph of distance from a fixed point against time, the rate describes how that distance changes as time passes. A positive rate means the distance increases. A negative rate means it decreases. A zero rate means it stays the same.

Slope is a measure of a graph segment’s steepness. Over an interval, the rate of change can be represented by the slope between two points. A larger positive rate gives a steeper upward segment. A more negative rate gives a steeper downward segment. These ideas describe a graph’s direction and steepness without requiring an equation.

The rate equals the change in the vertical quantity divided by the change in the horizontal quantity. The symbol

\Delta

means “change in.” In the expression,

r

is the rate,

\Delta y

is the change in the vertical quantity, and

\Delta x

is the change in the horizontal quantity.

r=\frac{\Delta y}{\Delta x}

Read a graph from left to right.
A positive rate means the graph rises; a negative rate means it falls; a zero rate means it is horizontal.
A greater rate size means a steeper graph when both axes use the same scales.

2. Translate rate words into graph shape

First identify the intervals in the description. Words such as “then,” “for the next few minutes,” and “until” can signal that the behaviour changes. Mark those points in order along the horizontal axis.

Next decide whether the rate is positive, negative, or zero on each interval. Its sign tells you whether the graph rises, falls, or stays horizontal.

Then consider whether the rate’s size changes. If a positive rate gets larger, the graph keeps rising and becomes steeper. If a positive rate gets smaller, the graph still rises but becomes less steep. If a negative rate becomes more negative, the graph falls more steeply. If a negative rate moves closer to zero, the graph keeps falling but becomes less steep.

“At a constant rate” means the rate stays the same over an interval. A straight segment represents a constant rate. “At an increasing rate” means the rate is getting larger. By itself, this does not say whether the quantity is rising or falling. For example, a quantity can decrease at an increasing rate: its graph falls and becomes steeper.

A verbal description may not provide exact values or a scale. In that case, your sketch should show the correct direction and changes in steepness. Do not suggest that approximate positions are exact coordinates.

The rate’s sign tells you whether the graph rises, falls, or stays level.
The rate’s size tells you how steep the graph is.
Separate the quantity’s direction from how its rate changes.

3. Plan and sketch interval by interval

A useful method is to label the axes and units, mark the intervals, and decide the rate’s sign on each one. Then decide whether the steepness stays the same, increases, or decreases. Sketch the intervals in order.

Suppose a cyclist’s distance from a starting point is recorded over time. At first, the cyclist travels away at a steady pace. The distance rises at a constant rate, so draw a straight rising segment. If the cyclist then speeds up while continuing away, the distance still rises, but its rate of increase grows. Draw a segment that becomes steeper upward. If the cyclist stops, the distance stays the same, so draw a horizontal segment.

This example shows why “distance increases” is not enough to determine the full shape. That phrase tells you the graph rises. “At a steady pace” means it is straight, while “speeds up” means its steepness changes.

Join intervals in a way that matches the description. A gradual rate change can be shown with a gradual change in steepness. A sudden rate change can be shown with a corner. If the wording does not say whether a change is gradual or sudden, make a reasonable sketch and avoid claiming more detail than the description gives.

A steady rate gives a straight segment.
A changing rate means the graph’s steepness changes.
A constant quantity gives a horizontal segment.

4. Use a table as a planning tool

A small table can help organize a verbal description before you draw. For each interval, record the rate’s sign and whether its size stays the same, increases, or decreases. The sign gives the graph’s direction. The rate’s size helps determine its steepness.

For instance, a positive rate that becomes larger gives a rising graph that grows steeper. A negative rate that moves closer to zero gives a falling graph that becomes less steep. A zero rate gives a horizontal segment.

Use the table to plan the shape, not to invent exact plotted points. When no numerical times or values are provided, the goal is a sketch that communicates the described behaviour.

Record rate direction separately from rate size.
Plan each interval before drawing the full graph.
Do not invent exact values when the description provides none.

From rate description to graph behaviour

Rate description	Graph behaviour
Positive and constant	Straight segment rising to the right
Positive and getting larger	Rises and becomes steeper
Positive and getting smaller	Rises and becomes less steep
Zero	Horizontal segment
Negative and constant	Straight segment falling to the right
Negative and moving toward zero	Falls and becomes less steep
Negative and becoming more negative	Falls and becomes steeper

Worked example

A container filling and draining

Sketch a graph of the amount of water in a container against time. The container starts empty. Water enters at a constant rate for a while. The inflow then gradually slows, but water continues to enter. Next, the amount stays constant for a short time. Finally, water drains out at a constant rate until the container is empty.

Set up the axes
Put time, $t$ , on the horizontal axis and amount of water, $W$ , on the vertical axis. Both quantities are nonnegative. Mark, in order, the start, the change to slower filling, the pause, and the end of draining. No exact times are given, so the intervals do not need numerical labels.
Draw the first interval
The amount starts at zero. Water enters at a constant positive rate, so the amount increases steadily. Represent this with a straight rising segment beginning at the origin.
$W(0)=0$
Show the slowing inflow
Water continues to enter, so the amount keeps rising. The rate of increase gets smaller, so the graph becomes less steep. It does not turn downward because the amount is still increasing.
$r_1>r_2>0$
Show the pause
The amount stays constant for a short time. Draw a horizontal segment above zero because the container already contains water.
$r=0$
Show the draining interval
The amount decreases while water drains. A constant draining rate gives a straight falling segment. End the segment at zero, because the container is empty then.
$W=0$

Answer: The sketch starts at zero, rises in a straight line, continues rising while becoming less steep, stays horizontal briefly, and then falls in a straight line to zero. Time is on the horizontal axis; amount of water is on the vertical axis.

Check: The sketch shows constant filling, slowing filling, no change, and constant draining in the stated order. It does not show the amount becoming negative.

Common mistakes and how to avoid them

Drawing a falling graph whenever the rate is described as decreasing.

Correction: A decreasing rate does not always mean the quantity decreases. Check the rate’s sign first. A positive rate that gets smaller still gives a rising graph; it simply becomes less steep.

Drawing a rising graph when a quantity decreases at an increasing rate.

Correction: “Increasing rate” describes how the rate changes, not necessarily whether the quantity rises. If the rate is negative and becomes more negative, the graph falls more steeply.

Using a curved graph for every changing quantity.

Correction: Use the rate description to decide the shape. A constant rate gives a straight segment. A changing rate calls for changing steepness; the sketch need not use a particular curve.

Adding exact values that the verbal description does not provide.

Correction: Label the quantities and show the required shape. Choose numerical values only when the description supports them.

Lesson summary

Label the axes and identify which quantity depends on the other.
Use the rate’s sign to decide whether the graph rises, falls, or stays level.
Use the rate’s size and how it changes to decide the graph’s steepness.
Mark where the description changes, then sketch each interval in order.
Match the description without implying unsupported exact values.

Check your understanding

Question 1

A plant’s height increases, but its rate of growth becomes smaller over several weeks. What should the graph do over that interval?

Rise and become less steep
Fall and become steeper
Stay horizontal
Rise at a constant steepness

Show answer and explanation

Rise and become less steep

The height continues to increase, so the graph rises. The positive rate becomes smaller, so the graph becomes less steep.

Question 2

A car’s distance from a fixed point decreases at a constant rate. Which graph segment fits?

A straight segment falling to the right
A straight segment rising to the right
A horizontal segment
A rising segment that becomes steeper

Show answer and explanation

A straight segment falling to the right

The distance decreases, so the rate is negative. A constant rate is represented by a straight segment.

Question 3

A quantity is falling, and its rate becomes less negative. What happens to the graph’s steepness?

It continues falling but becomes less steep
It turns upward immediately
It falls more steeply
It becomes horizontal at once

Show answer and explanation

It continues falling but becomes less steep

The rate remains negative, so the quantity continues to fall. Moving closer to zero means the downward steepness decreases.

Key terms

Independent variable: The input quantity shown on the horizontal axis.
Dependent variable: The quantity whose value depends on the input, usually shown on the vertical axis.
Rate of change: A comparison of how much one quantity changes for a change in another quantity.
Slope: A measure of a graph segment’s steepness, found by comparing vertical change with horizontal change.
Constant rate: A rate that stays the same over an interval.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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