In scientific experiments and mathematical modelling, data collected for two related variables often does not display an immediate straight-line relationship when plotted in their original forms. This topic introduces the distinction between linear and non-linear relations, the technique of drawing a line of best fit, determining its equation in the linear form \(Y = mX + c\), and interpreting values through interpolation and extrapolation.
Identify the correct distribution of points on both sides of a line of best fit and understand how the gradient \(m\) and \(Y\)-intercept \(c\) relate to the linear equation \(Y = mX + c\).
Points pass through several coordinates and stray points are balanced equally above and below the line.
All stray points lie on only one side of the line, so the line does not represent the data fairly.
The vertical axis is \(Y\), the horizontal axis is \(X\), the intercept is \(c\) at \(X = 0\), and the gradient is \(m = \dfrac{\Delta Y}{\Delta X}\).
A relationship is linear if plotting one variable against another produces a straight line. If the plotted points form a curve, the relationship is non-linear.
When plotted as a graph, data points form a straight line. It fits the general linear form \(Y = mX + c\), where \(X\) is the horizontal-axis variable, \(Y\) is the vertical-axis variable, \(m\) is the gradient, and \(c\) is the \(Y\)-intercept.
When plotted as a graph, data points form a curve rather than a straight line. A linear graph can be obtained when the variables plotted on the horizontal axis, vertical axis, or both are changed.
For \(y = x^2 + 1\), plotting \(y\) against \(x\) yields a curve, whereas plotting \(y\) against \(x^2\) yields a straight line.
A line of best fit is a straight line that joins most of the points plotted on the graph. Points that do not lie directly on the line must be distributed evenly on both sides of the straight line.
Draw the line of best fit through the plotted points.
Choose two points \((X_1, Y_1)\) and \((X_2, Y_2)\) on the line.
Calculate the gradient \(m = \dfrac{Y_2 - Y_1}{X_2 - X_1}\).
Read or calculate the \(Y\)-intercept \(c\), then write \(Y = mX + c\).
Gradient of a straight line
Use two points that lie on the drawn line of best fit.
General linear form
\(m\) is the gradient and \(c\) is the \(Y\)-intercept.
Answer in your mind, then press "Check Answer".
A straight line graph is obtained by plotting \(\dfrac{1}{y}\) against \(\dfrac{1}{x}\). The line passes through the points \((1, 7)\) and \((19, 20)\). Find the gradient of the line.
State the condition that must be fulfilled by points that do not lie on the line of best fit.
A straight-line graph of \(\log_{10} y\) against \(x\) has a gradient of \(2\) and a vertical intercept of \(0.2\). Express the equation of the line of best fit in the form \(Y = mX + c\).
A non-linear relation can never be represented by a straight-line graph.
A non-linear equation can produce a straight-line graph if the variables on the axes are appropriately transformed, for example plotting \(Y = y\) against \(X = x^2\) for \(y = x^2 + 1\).
The line of best fit must pass through every plotted experimental point.
Experimental data usually contains variations. The line of best fit needs to pass through as many points as possible while balancing the remaining points evenly on either side.
The line of best fit must always pass through the origin \((0, 0)\).
The line only passes through the origin if the \(Y\)-intercept \(c = 0\). In many relations, \(c \neq 0\).
Objective: Identify linear and non-linear relations, lines of best fit, gradients, intercepts and key terms in linear law.
1 What is the primary defining characteristic of the points that do not lie directly on a drawn line of best fit?
2 When plotting the equation \(y = x^2 + 1\), which graph produces a linear relation?
3 For a linear graph representing \(Y = mX + c\), \(m\) represents the ________ of the line.
4 The intersection point of a straight line with the vertical axis is known as the ________.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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