Linear graphs: rate and starting value
Read the meaning of a gradient and an intercept from a simple model.
Before you begin
- Coordinate pairs
- Solving two-step equations
Interpret the gradient
In y = mx + c, m is the change in y for each unit increase in x. Calculate it as the change in y divided by the change in x between two points. Keep the coordinate order consistent in both subtractions. A negative gradient means y decreases as x increases. Its units are the y-axis units divided by the x-axis units.
Interpret the intercept in context
The intercept c is the value of y when x is zero. It may represent a starting amount or a fixed charge. A straight-line model assumes a constant rate over the interval used. Extending a graph far beyond the available data can be unreasonable, even if the algebra still produces a number.
Worked example
A hire charge is C = 4h + 12 dollars for h hours. Find the cost for 3 hours.
Show the worked solution
- The fixed charge is $12 and the hourly rate is $4.
- Substitute h = 3: C = 4 × 3 + 12.
- The variable charge is $12; add the fixed $12.
$24.
Try it yourself
What is the gradient of the line through (1, 5) and (3, 11)?
Common mistakes
- The intercept is not generally the y-coordinate of any given point.
What can you explain now?
Find the equation of a line with gradient 2 that passes through (3, 10).
Compare with the explanation
y = 2x + 4.
Substitute the point into y = 2x + c: 10 = 6 + c, so c = 4. Check that x = 3 gives y = 10.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗