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Year 8AC9M8A03

Modelling and Graphing Linear Relations

Use mathematical modelling to solve applied problems involving linear relations, including financial contexts. Experiment with linear functions and relations using digital tools, making and testing conjectures and generalising emerging patterns.

A real situation, a rule, a table and a graph are four views of the same thing. Word problems get easy once you can move between them.

Builds onYr 8 · Linear Equations and Inequalities

Method

Write the rule from the words
cost=fixed+rate×number\text{cost} = \text{fixed} + \text{rate} \times \text{number}
The fixed part is the $y$-intercept. The rate is the gradient.
The gradient is how fast it changes
m=riserunm = \frac{\text{rise}}{\text{run}}
Per one unit across.
The intercept is the starting value
cc
Where the line crosses the $y$-axis.
Test the model against reality
If the answer is impossible, the model is wrong, not the arithmetic.

Worked example

e.g. A plumber charges $60 call-out plus $45 per hour. Write the rule and find the cost of 4 hours.\text{A plumber charges } \$60 \text{ call-out plus } \$45 \text{ per hour. Write the rule and find the cost of 4 hours.}
  1. Identify the fixed part. It happens once.6060
  2. Identify the rate. It happens per hour.45 per hour45 \text{ per hour}
  3. Write the rule.C=60+45hC = 60 + 45h
  4. Substitute 4 hours.C=60+45(4)C = 60 + 45(4)
  5. Work it out.C=$240C = \$240
  6. Read the graph meaning: intercept 60, gradient 45.
    The cost line with the four hour point markedxyC = 60 + 45h(4, 240)

Practice

Find the fixed part and the rate, then write the rule.

1
Gym: $30 join fee+$12/week. Write the rule.\text{Gym: } \$30 \text{ join fee} + \$12/\text{week. Write the rule.}
AnswerC=30+12wC = 30 + 12w
2
Using that rule, find the cost of 8 weeks.\text{Using that rule, find the cost of 8 weeks.}
Answer$126\$126
3
A line has gradient 3 and intercept −2. Write it.\text{A line has gradient 3 and intercept } -2. \text{ Write it.}
Answery=3x−2y = 3x - 2
4
In y=5x+20, what does the 20 mean?\text{In } y = 5x + 20, \text{ what does the 20 mean?}
AnswerThe starting value, before any x.\text{The starting value, before any } x.
5
A taxi costs $5+$2,20/km. Cost of 12 km?\text{A taxi costs } \$5 + \$2{,}20/\text{km. Cost of } 12 \text{ km?}
Answer$31,40\$31{,}40
6
Your model gives a negative cost. What does that mean?\text{Your model gives a negative cost. What does that mean?}
AnswerThe model does not apply there. Check the range.\text{The model does not apply there. Check the range.}
Next step
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