Turning curves into straight lines Cambridge IGCSE Additional Mathematics revision
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In plain words
A curve drawn through experimental points is hard to judge. A straight line is easy: you can see at once if the points fit, and read off a gradient and an intercept. So relationships that are really curves are redrawn, using logs or new axes, as straight lines.
6 things to know
- A straight line has the form Y = mX + c, where m is the gradient and c is the intercept on the vertical axis.
- y = Axⁿ: take logs to get lg y = n lg x + lg A. Plot lg y against lg x. The gradient is n and the intercept is lg A.
- y = Abˣ: take logs to get lg y = x lg b + lg A. Plot lg y against x. The gradient is lg b and the intercept is lg A.
- Natural logs can be used in the same way, with ln in place of lg.
- Other forms: y = ax² + b is a straight line if y is plotted against x². The gradient is a and the intercept is b.
- Going the other way: find the gradient and intercept of the line, write its equation using what is on each axis, then rearrange to get y in terms of x.
Worked example
Variables x and y are related by y = Abˣ. The graph of ln y against x is a straight line through (0, 1.2) and (4, 3.6). Find A and b.
- Taking logs: ln y = x ln b + ln A. So the gradient is ln b and the intercept is ln A.
- Gradient = (3.6 − 1.2) ÷ (4 − 0) = 0.6. So ln b = 0.6 and b = e^0.6 = 1.82.
- Intercept = 1.2. So ln A = 1.2 and A = e^1.2 = 3.32.
Tips and tricks
- Write the equation of the line with what is on the axes, such as ln y = 0.6x + 1.2, before trying to find A and b.
- The intercept gives the log of A, not A itself. Undo the log as a last step.
It lands in your notebook with its questions as flashcards.
Turning curves into straight lines: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
y = Axⁿ. Which graph is a straight line?
lg y = n lg x + lg A.
y = Abˣ. Which graph is a straight line?
lg y = x lg b + lg A.
For y = Axⁿ plotted as lg y against lg x, what does the gradient give?
The intercept gives lg A.
A graph of lg y against lg x has intercept 2. What is A in y = Axⁿ?
lg A = 2, so A = 10².
y = ax² + b. What should be plotted to get a straight line?
Then the gradient is a and the intercept is b.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
Turning curves into straight lines
Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org
Name ______________________________ Date ______________
State what should be plotted to obtain a straight line from y = Axⁿ.[2]
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lg y against lg x. (Natural logs can be used in the same way.)
The graph of y against x² is a straight line through (1, 5) and (4, 11). Find y in terms of x.[3]
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y = 2x² + 3. The gradient is 6/3 = 2, and 5 = 2 × 1 + c gives c = 3.
The graph of ln y against x is a straight line with gradient 0.5 and intercept 2. Given that y = Abˣ, find A and b to 3 significant figures.[3]
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A = 7.39 and b = 1.65. A = e² and b = e^0.5.
Answers: Turning curves into straight lines
- 1. lg y against lg x. (Natural logs can be used in the same way.)
- 2. y = 2x² + 3. The gradient is 6/3 = 2, and 5 = 2 × 1 + c gives c = 3.
- 3. A = 7.39 and b = 1.65. A = e² and b = e^0.5.