178e. Plotting Non-Linear Relationships

Learning Intentions

• Recognise that some rules relating and can result in graphs where the points do not lie on a line
• Plot a non-linear relationship by creating a table of values

Pre-requisite Summary

• A relationship between and can be shown Use a table of values or a Draw
• Ordered pairs are written as
• A point lies on a graph if its coordinates satisfy the rule
• A linear relationship has a constant rate of change and its graph is a straight line
• Substitution can be used to Solve matching values of for chosen values of
• A graph can be constructed by plotting points from a table of values

Worked Examples

Worked Example 1

For the rule , Draw a table of values for .

Worked Example 2

For the rule , plot the points from the table and Describe whether they lie on a straight line.

Worked Example 3

For the rule , construct a table of values for and plot the graph.

Worked Example 4

Determine whether the rule is linear or non-linear by comparing the change in values as increases by .

Worked Example 5

For the rule , construct a table of values for and plot the graph.

Worked Example 6

A student says that any graph made from a table of values must be a straight line.

Use the rule to test this claim.

Problems

Problem 1

For the rule , construct a table of values for .

Problem 2

For the rule , plot the points from the table and describe whether they lie on a straight line.

Problem 3

For the rule , construct a table of values for and plot the graph.

Problem 4

Determine whether the rule is linear or non-linear by comparing the change in values as increases by .

Problem 5

For the rule , construct a table of values for and plot the graph.

Problem 6

A student says that the rule should form a straight line because there is an equation.

Use a table of values to test this claim.

Exercises

Understanding and Fluency

Exercise 1.

Complete each table of values.

a) For , when

b) For , when

Exercise 2.

Complete each table of values.

a) For , when

b) For , when

Exercise 3.

Plot the Points from Each Rule.

a)

b)

Exercise 4.

Plot the Points from Each Rule.

a)

b)

Exercise 5

For each rule, State whether the graph is linear or non-linear.

a)

b)

c)

Exercise 6

For each rule, state whether the graph is linear or non-linear.

a)

b)

c)

Exercise 7

Construct a table of values for each rule.

a) for

b) for

Exercise 8

Construct a table of values for each rule.

a) for

b) for

Exercise 9

Compare the change in values as increases by .

a)

b)

State which rule is linear and which is non-linear.

Exercise 10

A rule has the table of values below.

a) Plot the points

b) Decide whether the graph is a straight line

c) State whether the relationship is linear or non-linear

Reasoning

Exercise 11

Explain why the graph of is not a straight line.

Exercise 12

A student says that if increases by , then must also increase by a constant amount. Explain why this is true for linear rules but not for all rules.

Exercise 13

Explain why a table of values is useful when plotting a non-linear relationship.

Exercise 14

A student plots the points for and joins them with straight line segments only. Explain why this does not show the relationship accurately.

Exercise 15

Explain why the points for are symmetric about the -axis.

Problem-solving

Exercise 16

A pattern follows the rule , where is the step number.

Construct a table of values for , then plot the points.

Exercise 17

The area of a square with side length cm is given by .

Construct a table of values for , then plot the relationship.

Exercise 18

A student claims that the rule is linear because it has a plus sign.

Construct a table of values and use it to test the claim.

Exercise 20

A graph is made from the rule .

Construct a table of values for , then state whether the graph is linear or non-linear.

Exercise 21

The points , , , and are given.

Construct a graph from the table of points and describe the shape formed.

Potential Misunderstandings

• Thinking every rule produces a straight-line graph
• Believing that any relationship shown in a table must be linear
• Forgetting to Substitute carefully when calculating values such as
• Confusing with
• Thinking a non-linear graph cannot be drawn from a table of values
• Expecting the change in to be constant for every rule
• Plotting the points correctly but then joining them as if they lie on a straight line
• Forgetting that a non-linear graph can still show a clear pattern
• Assuming that a plus sign in a rule always means the graph is linear