In scientific experiments and real-life mathematical applications, variables often have non-linear relationships such as quadratic, exponential, or inverse relationships. By applying linear law, these non-linear equations are transformed into the linear form \(Y = mX + c\). A line of best fit is plotted using experimental data to determine unknown constants from the gradient and the vertical intercept, identify incorrectly recorded data points, and make estimations or predictions.
Follow the systematic path from experimental data to calculating unknown physical constants.
Understanding the problemIdentify given variables, constants, the non-linear equation, required axes and scales.
Planning the strategyReduce the equation to \(Y = mX + c\), identify \(Y\), \(X\), \(m\) and \(c\), then construct a new table of values.
Implementing the strategyPlot points, draw the line of best fit, calculate \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) and read \(c\) where \(X = 0\).
Making a conclusion / checkingEquate \(m\) and \(c\) to algebraic expressions to solve for unknown constants, then verify with the original equation.
General linear form
Gradient \(m\) and \(Y\)-intercept \(c\) determine the unknown constants.
Gradient from two points on the line of best fit
Choose points that lie precisely on the line, not raw experimental points.
Transform non-linear models into linear form, then use the line of best fit to find unknown constants.
Applying linear law involves taking non-linear mathematical models that arise in biological, physical, or commercial experiments and manipulating them algebraically into the standard linear equation \(Y = mX + c\).
Each data pair \((x, y)\) is converted into new coordinates \((X, Y)\) using the transformed variables.
A graph of \(Y\) against \(X\) is plotted, and a single straight line of best fit is drawn such that points are evenly distributed above and below it.
The gradient \(m\) is calculated from two points on the line (not necessarily experimental points), and the intercept \(c\) is read where \(X = 0\).
By comparing the algebraic linear equation to \(Y = mX + c\), the unknown constants are directly computed.
If one data point diverges significantly from the linear trend, it represents an experimental error (marked \(\otimes\)), and its true value can be restored using the line of best fit.
The resulting linear graph or converted equation can be used to interpolate or extrapolate values.
Exponential form \(y = pq^x\)
\(q = 10^m\) and \(p = 10^c\).
Lens equation \(\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}\)
Gradient \(m = -1\) and \(f = \dfrac{1}{c}\).
Answer in your mind, then press "Check Answer".
Given the non-linear equation \(y\sqrt{x} = 10\), a student plots a straight-line graph of \(y\) against \(\dfrac{1}{\sqrt{x}}\). The line passes through the origin \((0, 0)\) and the point \((2, p)\). Find the value of \(p\).
The variables \(P\) and \(K\) are related by \(P = 180 + rK - sK^2\). When \(\dfrac{P - 180}{K}\) is plotted against \(K\), the resulting line of best fit has a \(Y\)-intercept of \(1.6\) and passes through the point \((10, -4.00)\). Find the values of \(r\) and \(s\).
Variables \(u\) and \(v\) satisfy the relation \(\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}\). When a graph of \(\dfrac{1}{v}\) against \(\dfrac{1}{u}\) is plotted, the line of best fit has a vertical intercept of \(0.05\). State the gradient of the graph and determine the value of \(f\).
The line of best fit must pass through every plotted experimental point.
Experimental measurements often include errors. The line of best fit represents the general linear trend and should have an approximately equal balance of points above and below the line; it does not need to pass through all points.
The \(Y\)-intercept can be read directly from the vertical axis regardless of where the horizontal axis starts.
The \(Y\)-intercept is strictly the value of \(Y\) when the horizontal variable \(X = 0\). If the graph starts with a broken axis or an initial value other than \(0\), the intercept cannot be read directly from the line and must be calculated using \(c = Y - mX\).
An outlying point is just a normal variation and should be forced into the line of best fit.
A point that deviates excessively from the trend represents a recording or experimental error. It should be identified, marked with \(\otimes\), and excluded when positioning the line of best fit.
Objective: Test your understanding of linear law transformations, gradients and intercepts.
1 Variables \(x\) and \(y\) are related by the equation \(y\sqrt{x} = 10\). Diagram (a) shows a straight-line graph plotted as \(y\sqrt{x}\) against \(x\), which is a horizontal line with vertical intercept \(p\). What is the value of \(p\)?
2 The variables \(t\) and \(p\) are related by the formula \(\sqrt{p} = \dfrac{t}{k}\). When a graph of \(t^2\) against \(p\) is plotted, which expression represents the gradient of the straight line?
3 In an experiment where an erroneous data point deviates noticeably from the line of best fit, the incorrect point is marked with the symbol ________ on the graph.
4 When the equation \(y = pq^x\) is converted to linear form by taking base-10 logarithms, the gradient of the graph of \(\log_{10} y\) against \(x\) is equal to ________.
Match each non-linear equation and its plotted axes to the correct gradient expression. On a phone, tap a term first, then tap the matching box.
There is something wrong with this question.
