Additional Mathematics – Straight-line graphs | e-Consult
Straight-line graphs (1 questions)
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Answer 2
Take common logarithms of both sides:
\(\log y = \log k + n\log x\)
Let \(Y = \log y\) and \(X = \log x\); the plot of \(Y\) against \(X\) is a straight line with gradient \(n\) and intercept \(\log k\).
| x | y | \(\log x\) | \(\log y\) |
|---|---|---|---|
| 1 | 3 | 0.000 | 0.4771 |
| 2 | 12 | 0.3010 | 1.0792 |
| 4 | 48 | 0.6021 | 1.6812 |
| 8 | 192 | 0.9031 | 2.2833 |
Using any two points (e.g., the first two) the gradient is
\(n = \dfrac{1.0792-0.4771}{0.3010-0.0000}= \dfrac{0.6021}{0.3010}=2.00\)
Intercept:
\(\log k = Y - nX = 0.4771 - 2(0) = 0.4771\)
Hence \(k = 10^{0.4771}=3.00\).
Final relationship (to three significant figures):
\(y = 3.00\,x^{2.00}\)