Worked example 1
Determine an exponential model
For y = abx, y = 6 when x = 0 and y = 24 when x = 2. Find a and b, where b > 0.
- At x = 0, a = 6
- Use 24 = 6b².
- Solve b² = 4 and use b > 0.
Answer: y = 6(2x)
Higher Mathematics
Higher Maths Exponential modelling and linearisation
Practise exponential modelling and linearisation for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Model growth and decay, find constants from data and linearise power or exponential relationships using logarithms.
Before you start
- Use logarithm laws.
- Find a straight-line gradient and intercept.
- Solve equations involving powers.
Explanation
Exponential models y = abx describe repeated multiplication, while power models y = axb describe scaling by a power.
Taking logarithms produces a straight-line form: log y = log a + x log b for an exponential model, or log y = log a + b log x for a power model.
Method and rules
- If y = abx, then log y = log a + x log b.
- If y = axb, then log y = log a + b log x.
- In a log y against log x graph, gradient = b and intercept = log a
Worked examples
Worked example 2
Method check
For y = axb, what is the gradient of a log y against log x graph?
- Compare with Y = mX + c
So: b.
Watch out
- Using the untransformed variables as the straight-line axes.
- Confusing log a with a.
- Accepting a negative base in a real exponential model.
Exam reminder
For exponential modelling and linearisation, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.