Lesson 1 of 6Linear and Non-Linear Relations

Unit 5 · Lesson 1 of 6

Linear and Non-Linear Relations

Use tables of values, first differences and the shape of a graph to decide whether a relation is linear or non-linear, and describe growing and shrinking patterns.

8 min read7 quick checks11 practice questionsAsk AI about this →

Key things to remember

1A linear relation has a straight-line graph; in a table with equal $x$ -steps, its first differences are all equal.
2First difference = next $y$ -value $-$ current $y$ -value, found between consecutive rows.
3With equal $x$ -steps, first differences that change mean the relation is non-linear and the graph curves.
4A positive constant first difference gives a rising line, a negative one a falling line, and $0$ a horizontal line.
5If the $x$ -steps are unequal, compare $\frac{change in y}{change in x}$ for each pair of rows instead.

You start a summer job and save 25 dollars every week. After 1, 2, 3 and 4 weeks you have 25, 50, 75 and 100 dollars: the total goes up by the same amount each week. Meanwhile a friend posts a video that gets 100 views on day 1, and the number of views doubles every day: 100, 200, 400, 800, 1600. Here the jumps get bigger and bigger. Both situations connect two quantities (time and money, time and views), so both are relations. The first is a linear relation, and its graph is a straight line. The second is non-linear, and its graph curves upward. In this lesson you will learn a quick test, called first differences, that tells the two kinds apart from a table of values, and you will see why the difference matters when you try to predict what happens next.

Relations, variables and tables of values

A relation is a connection between two variables. You can describe it in words, with a table of values, with a graph, or with an equation. The independent variable is the one you choose or that changes on its own, such as time or the stage number of a pattern. It is usually called $x$ and goes on the horizontal axis. The dependent variable is the one that depends on it, such as money saved or number of tiles. It is usually called $y$ and goes on the vertical axis. A table of values lists matching pairs: each row is one ordered pair $(x, y)$ and one point on the graph.

Fig 1A table of savings of 25 dollars a week beside its graph: weeks 1 to 4 give the points (1, 25), (2, 50), (3, 75) and (4, 100), with the row for week 3 traced to its point.

Quick checkTry it before you read on

The height of a sunflower is measured every week. Which is the dependent variable?

Quick checkTry it before you read on

A table of values has a row with

x = 4

and

y = 13

. Which point on the graph does that row give?

First differences

When the $x$ -values in a table go up by equal steps (most often by 1), subtract each $y$ -value from the next one. These results are the first differences. If every first difference is the same number, the relation is linear: each equal step to the right gives the same rise or fall, so the points line up in a straight line. If the first differences are not all the same, the relation is non-linear and its graph curves. For $y = 3 x + 2$ the $y$ -values $2, 5, 8, 11, 14$ have first differences $+ 3, + 3, + 3, + 3$ . For $y = x^{2} + 1$ the $y$ -values $1, 2, 5, 10, 17$ have first differences $+ 1, + 3, + 5, + 7$ , which keep changing.

first difference = (next y -value) - (current y -value)

Fig 2Tables of values for y = 3x + 2 and y = x squared + 1 with their first differences, each beside its graph showing the rise at every step

Check the x-steps first

First differences only work directly when

x

goes up by equal steps. The table

x

0, 1, 3, 4

with

y

1, 3, 7, 9

comes from

y = 2 x + 1

, a linear relation, yet its first differences are

2, 4, 2

. The middle jump is bigger only because

x

jumped by 2 there. When the

x

-steps are uneven, divide each change in

y

by the matching change in

x

\frac{2}{1}, \frac{4}{2}, \frac{2}{1}

all equal

2

, so the relation is linear.

Quick checkTry it before you read on

A table has

x

0, 1, 2, 3

and

y

5, 3, 1, - 1

. What is the first difference?

Quick checkTry it before you read on

A table has

x

0, 2, 4, 6

and

y

1, 7, 13, 19

. What can you conclude?

VideoAn Ontario teacher shows how to subtract consecutive y-values in a table and use constant first differences to decide whether a relation is linear.Watch on YouTube

Shapes of graphs

What the table shows (equal x-steps)	Type of relation	Shape of the graph
Same positive first difference, e.g. +4, +4, +4	Linear, growing	Straight line rising from left to right
Same negative first difference, e.g. −3, −3, −3	Linear, shrinking	Straight line falling from left to right
First differences all 0, e.g. 6, 6, 6, 6	Linear, constant	Horizontal straight line
First differences change, e.g. +1, +3, +5, +7	Non-linear	A curve
Each y is the previous y times 2, e.g. 3, 6, 12, 24	Non-linear, growing faster and faster	A curve that gets steeper
Each y is half the previous y, e.g. 64, 32, 16, 8	Non-linear, shrinking more and more slowly	A curve that flattens out

Reading the type of relation from a table of values

Fig 3Six tables plotted point by point: constant differences of +4, −3 and 0 give a rising, a falling and a horizontal line, while changing differences, doubling and halving give curves. Each graph has its own vertical scale.

Quick checkTry it before you read on

With equal

x

-steps, a table's

y

-values are

40, 20, 10, 5

. What does its graph look like?

Growing and shrinking patterns

Patterns can grow or shrink, and either way they can be linear or not. A tile pattern with 5, 9, 13, 17 tiles in stages 1 to 4 grows by 4 tiles each stage: linear and growing. A 24 cm candle that burns down 3 cm every hour has heights 24, 21, 18, 15: first differences of $- 3$ , so it is linear and shrinking. A square patio that is 1, 2, 3, 4 tiles wide needs 1, 4, 9, 16 tiles: first differences 3, 5, 7, so it is non-linear. A ball dropped from 160 cm that bounces back to half its height each time reaches 80, 40, 20, 10 cm: first differences $- 40, - 20, - 10$ , so it is non-linear and shrinking. One more detail: a pattern with stage numbers only has whole-number stages (there is no stage 2.5), so you plot separate points. Time, distance and volume can take any value in between, so for them you join the points with a line or curve.

ALinear relation

Graph is a straight line
Equal x-steps give equal first differences
Changes by adding the same amount each step
Equation looks like $y = 3 x + 2$ (no $x^{2}$ , no $x$ in an exponent)

BNon-linear relation

Graph is a curve
Equal x-steps give first differences that change
Change speeds up or slows down
Equations such as $y = x^{2} + 1$ or $y = 100 \times 2^{x}$

Words that hint at the type

Phrases like "25 dollars per week" or "burns 3 cm each hour" (the same amount added or subtracted every step) usually signal a linear relation. Phrases like "doubles every day", "halves each bounce" or "the area of a square of side

x

" usually signal a non-linear one. Always confirm with a table: the hint is a starting guess, not a proof.

Why does this matter? For a linear relation you can predict by adding the same first difference again and again: savings after 10 weeks are $25 \times 10 = 250$ dollars. For a non-linear relation that shortcut fails badly. The video's views go up by 100 from day 1 to day 2, but on day 10 the views are $100 \times 2^{9} = 51200$ , not the 1000 you would get by adding 100 per day. Deciding linear or non-linear first tells you which kind of rule to look for.

Quick checkTry it before you read on

A tile pattern has 2, 5, 8 and 11 tiles in stages 1 to 4. How should you show it on a graph?

Quick checkTry it before you read on

A pond's water level is 30, 28, 26 and 24 cm on days 1 to 4. Is this relation linear or non-linear?

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Is the relation in this table linear?

x

0, 1, 2, 3, 4

and

y

7, 11, 15, 19, 23

. If it is, predict

y

when

x = 5

Example 2

Is this relation linear or non-linear?

x

1, 2, 3, 4, 5

and

y

3, 6, 12, 24, 48

. Describe the pattern and predict

y

when

x = 6

Example 3

The

x

-values in this table are not evenly spaced:

x

0, 2, 3, 6

and

y

5, 11, 14, 23

. Is the relation linear?

Example 4

A shrinking tile pattern has 22 tiles in stage 1, 18 in stage 2 and 14 in stage 3. If it keeps shrinking the same way, in which stage will there be 2 tiles?

Where marks get lost

Using first differences when the $x$ -values do not go up by equal steps, and then calling a linear relation non-linear (as with $x$ : $0, 1, 3, 4$ and $y = 2 x + 1$ ).
Subtracting in the wrong order (current minus next). For $y$ : $20, 17, 14$ that gives $+ 3$ instead of $- 3$ and makes a shrinking pattern look like a growing one.
Thinking a decreasing relation cannot be linear. A constant negative first difference, like −3, −3, −3, is still linear: the line falls instead of rising.
Deciding from only two points. Any two points can be joined by a straight line, so you need at least three points (and every difference in the table) to judge.
Adding a first difference to predict far ahead in a non-linear pattern. The doubling pattern 100, 200, 400 is not 100 more each step, so adding 100 per step gives badly wrong predictions.

Flashcards 8

Relation

Independent variable

Dependent variable

First differences

+ 4 more

Practice

In each table,

x

goes

0, 1, 2, 3

. Which list of

y

-values comes from a linear relation?

A table has

x

0, 1, 2, 3

and

y

- 2, 3, 8, 13

. What is the first difference?

A linear relation has

x

0, 1, 2, 3

and

y

40, 34, 28, 22

. Continue the pattern: what is

y

when

x = 6

Every first difference in a table (with equal x-steps) is 0. What does the graph look like?

Put these steps in order for deciding whether a table of values shows a linear relation.

1Subtract each y-value from the next one to get the first differences
2Conclude: all equal means linear, not all equal means non-linear
3Compare the first differences with each other
4Check that the x-values go up by equal steps

In each table x goes 1, 2, 3, 4. Match each list of y-values to its description.

3, 7, 11, 15

50, 45, 40, 35

2, 4, 8, 16

64, 32, 16, 8

6, 6, 6, 6

A table has

x

1, 2, 3, 4

and

y

1, 4, 9, 16

. Is the relation linear or non-linear?

A table has

x

0, 1, 3, 6

and

y

2, 5, 11, 20

. Which statement is correct?

A tile pattern has 5, 9 and 13 tiles in stages 1, 2 and 3 and keeps growing the same way. How many tiles are in stage 10?

tiles

The number of views on a video doubles every day. Which graph shape fits views against days?

A candle is 24 cm tall and burns down 3 cm every hour. How tall is it after 5 hours?

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