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Year 9 Maths | Understand, justify, check

Read the rate and the starting value

A linear model combines a constant rate of change with a starting value. A graph, equation and context describe the same relationship, but each makes a different feature easy to see.

What to notice

In y = mx + b, m describes the change in y for each one-unit increase in x. The intercept b is the value of y when x is zero. It is not generally the rate.

What to do

Read both axis labels and scales. Find the starting value, then calculate change in y divided by change in x. Check the proposed equation against another point on the graph.

Why it works

Using two points tests the rate, while checking x = 0 tests the intercept. These checks separate a proportional relationship through the origin from a linear relationship with an initial charge.

Work through an example

A hire service charges $3 initially and $2 per hour. Model the cost.

  1. Let x be hours and y be total cost in dollars. Write y = 2x + 3.
  2. At zero hours, the model includes the $3 initial charge.
  3. At four hours, the model gives y = 2 × 4 + 3 = $11.

A common mistake

Reading the steepness by appearance alone is unreliable when graph scales differ. Calculate rise divided by run using the numbered axes.

A strategy to try

For two plans, write both expressions and set them equal to find a break-even point. Substitute that time into both models to confirm the costs match.

Try it now

Try three questions

No account is needed. These examples do not change saved learning progress. Choose an answer and check it for an explanation; you can try again.

1. The graph shows y = 2x + 1. What is y when x = 3?
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Graph of y = 2x + 1. Marked points: (0,1), (1,3), (2,5), (3,7), (4,9).
Help: a starting point

Locate x = 3 on the horizontal axis and move to the line.

Help: the next step

Check the reading by multiplying 3 by 2, then adding the intercept.

2. A line passes through (1, 4) and (4, 10). What is its gradient?
Help: a starting point

Find the change in y and the change in x between the two points.

Help: the next step

Divide vertical change by horizontal change, keeping that order.

3. Plan A costs $6 + $4 per hour. Plan B costs $14 + $2 per hour. After how many hours do they cost the same?
Help: a starting point

Write both total costs using the same variable for hours.

Help: the next step

Set the expressions equal and collect the hourly terms on one side.

For parents and carers

Ask the learner to explain what each coefficient means in the context. The graphs use explicit scales and labelled coordinates, while the cost comparison requires an equation. This is a sample of linear modelling, not a claim that every real cost follows a straight line.

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