Recover equations from transformed straight-line graphs
Choose transformed axes for a relationship and translate a measured gradient and intercept back into its original constants.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In Y = 5X − 2, what are the gradient and vertical intercept?
What you will learn
- Identify plotted variables that make a stated relationship linear
- Interpret the gradient and intercept for transformed equations
- Recover the original relationship without confusing axes or constants
Choose the plotted quantities from the model
If y² = Ax³ + B, plot Y = y² vertically against X = x³ horizontally. Then Y = AX + B, so the gradient is A and the vertical intercept is B.
If e^(2y) = Ax² + B, plot e^(2y) against x². The same gradient-intercept reading gives A and B.
If y³ = A ln x + B, plot y³ against ln x. The logarithm requires x > 0; the plotted gradient is A and the intercept is B.
Translate back only after reading the graph
Label the horizontal variable X and vertical variable Y before using Y = mX + c. Match m and c to the parameters in the transformed equation.
Then restore the original powers and functions exactly. For instance, a vertical intercept of −2 in a plot of e^(2y) against x² means B = −2, not y = −2.
Check by substituting the constants into the original relationship and comparing with the transformed straight-line equation.
Pause and explain
For y³ = A ln x + B, what is plotted on the horizontal axis?
Worked example
For y² = 3x³ + 4, state the axes and the gradient and intercept of the straight-line graph.
Show the worked solution
- Set Y = y² and X = x³.
- Then Y = 3X + 4.
- Compare with Y = mX + c to read m = 3 and c = 4.
Answer Plot y² against x³; gradient 3 and vertical intercept 4.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For e^(2y) = 5x² − 2, state the transformed axes, gradient and intercept.
Give me a hint
Set X = x² and Y = e^(2y).
Compare my reasoning
- Plot Y = e^(2y) vertically against X = x² horizontally.
- The equation becomes Y = 5X − 2.
- Read gradient 5 and vertical intercept −2.
Plot e^(2y) against x²; gradient 5 and intercept −2.
Look for these in your work
- Axes match variables
- Linear form identified
- Both constants read
A graph of y³ against ln x has gradient −2 and intercept 7. Write the original relationship.
Give me a hint
Match the graph to y³ = A ln x + B.
Compare my reasoning
- The gradient gives A = −2.
- The intercept gives B = 7.
- Restore the original variables: y³ = −2 ln x + 7, with x > 0.
y³ = −2 ln x + 7, for x > 0.
Look for these in your work
- Gradient assigned to A
- Intercept assigned to B
- Logarithm domain included
A straight-line plot of Y = y² against X = x³ has gradient 1/2 and intercept 9. State the relationship and check one transformed point: if x³ = 4, find y².
Give me a hint
Use Y = AX + B, then substitute X = 4.
Compare my reasoning
- A = 1/2 and B = 9.
- The original relationship is y² = (1/2)x³ + 9.
- At x³ = 4, y² = (1/2)(4) + 9 = 11.
y² = (1/2)x³ + 9; at x³ = 4, y² = 11.
Look for these in your work
- Constants recovered
- Original powers restored
- Transformed point checked
Common mistakes
- Reversing the horizontal and vertical plotted quantities
- Calling the intercept A when it represents B
- Dropping a square or cube while returning to the original variables
What can you explain now?
For e^(2y) = Ax² + B, which quantity is plotted vertically?
Compare with the explanation
e^(2y)
The equation is linear in X = x² and Y = e^(2y).
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Make it stick
How do you decide which quantities to plot and recover the constants from a transformed line?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.