177. Linear Graphs in Real-World Situations

Learning Intentions

• Understand that linear graphs can be applied to situations where there is a constant rate of change
• Apply linear graphs to model and Solve problems arising in real-world situations

Pre-requisite Summary

• A linear Draw represents a relationship that changes at a constant rate
• In a rule of the form , the gradient shows the rate of change
• The -intercept shows the starting value when
• Points on a graph represent solutions to the rule
• A table of values can be used to plot a linear graph
• The graph of a real-world situation should be interpreted Use the meaning of the variables and their units
• A solution found from a graph should be checked to see whether it makes sense in the context

Worked Examples

Worked Example 1

A taxi fare is given by the rule , where is the number of kilometres travelled and is the total cost in dollars.

Draw a table of values and graph the rule.

Worked Example 2

A water tank contains L of water and loses L each minute.

Write a linear rule, then graph the situation.

Worked Example 3

A plant starts at cm tall and grows cm each week.

Write a linear rule for the height of the plant, then graph the situation.

Worked Example 4

Two phone plans are modelled by and , where is the number of call units and is the total cost in dollars.

Graph both rules and Use the intersection point to Solve when the plans cost the same.

Worked Example 5

A bus is km from town and travels towards town at a constant speed of km/h.

Write a linear rule for the distance from town after hours, then graph the situation.

Worked Example 6

A student says that any real-world graph with a straight line must continue forever in both directions.

Use a context such as time or distance to test this claim.

Problems

Problem 1

A delivery charge is given by the rule , where is the number of kilograms and is the total cost in dollars.

Construct a table of values and graph the rule.

Problem 2

A tank contains L of water and loses L each minute.

Write a linear rule, then graph the situation.

Problem 3

A candle starts at cm and burns down by cm each hour.

Write a linear rule for the candle height, then graph the situation.

Problem 4

Two gym plans are modelled by and , where is the number of visits and is the total cost in dollars.

Graph both rules and use the intersection point to find when the plans cost the same.

Problem 5

A car is km from a city and travels towards the city at a constant speed of km/h.

Write a linear rule for the distance from the city after hours, then graph the situation.

Problem 6

A student says that the graph of a taxi fare rule should include negative values of distance.

Use the context to test this claim.

Exercises

Understanding and Fluency

Exercise 1

State whether each situation could be modelled by a linear graph.

a) A plant grows by cm each week

b) A balloon changes size unpredictably

c) A taxi fare increases by $2 per kilometre

Exercise 2

For each situation, identify the constant rate of change and the starting value.

a)

b)

c)

Exercise 3

Write a linear rule for each situation.

a) A phone starts with $ credit and gains $ each recharge unit

b) A tank starts with L and loses L each minute

c) A person is m from home and walks towards home at m/min

Exercise 4

Write a linear rule for each situation.

a) A candle is cm tall and burns cm each hour

b) A taxi fare has a $6 starting fee and costs $3 per kilometre

c) A plant is cm tall and grows cm each week

Exercise 5

Construct a table of values and graph each rule.

a)

b)

Exercise 6

Construct a table of values and graph each rule.

a)

b)

Exercise 7

Interpret each graph description.

a) A line crosses the -axis at and rises by for every across

b) A line crosses the -axis at and falls by for every across

For each, describe a possible real-world situation.

Exercise 8

Two rules are given by and .

a) Construct tables of values

b) Graph both rules

c) State the intersection point

Reasoning

Exercise 9

Explain why a constant rate of change leads to a straight-line graph.

Exercise 10

A student says that every real-world situation can be modelled by a linear graph. Explain why this is incorrect.

Exercise 11

Explain why the -intercept in a real-world graph often represents the starting amount.

Exercise 12

A student graphs a water tank model and continues the line into negative values of time. Explain why this may not make sense in context.

Exercise 13

Explain why the point of intersection of two linear graphs can solve a real-world problem.

Problem-solving

Exercise 14

A plumber charges a $15 call-out fee and $25 per hour.

Let be the number of hours and be the total cost.

a) Write a linear rule

b) Construct a table of values

c) Graph the rule

Exercise 15

A water tank starts with L and loses L each minute.

Let be the number of minutes and be the amount of water left.

a) Write the rule

b) Graph the rule

c) Use the graph to estimate when the tank will be empty

Exercise 16

Two payment plans are given by and , where is the number of uses.

a) Graph both rules

b) Find the intersection point

c) Explain what the intersection means in context

Exercise 17

A candle is cm tall and burns down by cm each hour.

a) Write a linear rule

b) Graph the situation

c) Use the graph to estimate the height after hours

Exercise 18

A bus is km from town and travels towards town at km/h.

a) Write a linear rule for the distance from town after hours

b) Graph the situation

c) Use the graph to estimate when the bus reaches town

Potential Misunderstandings

• Thinking any graph can model a constant rate of change
• Confusing the starting value with the rate of change
• Forgetting that a negative gradient can represent something decreasing at a constant rate
• Writing a rule that does not match the context or units of the variables
• Treating the graph as if it continues meaningfully into impossible values such as negative time or negative distance
• Forgetting that the intersection of two graphs represents equal values for the same input
• Reading a graph accurately in mathematics but interpreting it incorrectly in context
• Thinking a straight-line graph always means the value is increasing
• Ignoring whether the solution found from the graph makes sense in the real-world situation

Next: 178e. Plotting Non-Linear Relationships