Linear graphs: Practice Questions

The variables \(x\) and \(y\) satisfy the law \(y = ax^n\). When \(\log_{10} y\) is plotted against \(\log_{10} x\), a straight line is obtained. Which of the following represents the intercept of this line on the vertical axis?

A graph of \(y^2\) against \(x^3\) results in a straight line as shown in a coordinate plane. If the line passes through the points \((2, 7)\) and \((4, 13)\), find the values of the constants \(a\) and \(b\) in the relationship \(y^2 = ax^3 + b\).

A student is investigating the relationship \(\frac{1}{x} + \frac{1}{y} = k\), where \(k\) is a constant. They plot a graph of \(\frac{1}{y}\) on the vertical axis against \(\frac{1}{x}\) on the horizontal axis. What is the gradient of the resulting straight line?

The variables \(x\) and \(y\) satisfy the relationship \(y = \frac{a}{x} + b\). A diagram shows a straight line produced by plotting \(xy\) against \(x\). If the line passes through the points \((1, 5)\) and \((3, 11)\), find the value of \(a + b\).

A relationship between two variables \(x\) and \(y\) is given by the equation \(y = ab^x\), where \(a\) and \(b\) are constants. If a straight-line graph is to be drawn by plotting \(\log_{10} y\) against \(x\), what is the gradient of this line?

The variables \(x\) and \(y\) are related by the equation \(y^2 = ax^3 + b\). To represent this relationship as a linear graph, state the variables that should be plotted on the vertical and horizontal axes, and express the gradient of this line in terms of the constant \(a\).

The variables \(x\) and \(y\) satisfy the relationship \(y = ax^n\). When \(\log_{10} y\) is plotted against \(\log_{10} x\), a straight line with gradient \(3\) is formed which passes through the point \((2, 7)\). Calculate the exact values of the constants \(a\) and \(n\).

The variables \(x\) and \(y\) satisfy the equation \(y^2 = ax^3 + b\).

(a) Describe how a straight-line graph can be drawn to represent this relationship. State the variables that should be plotted on each axis.
(b) When this graph is drawn, it passes through the points \((8, 10)\) and \((27, 29)\), where the horizontal axis represents \(x^3\) and the vertical axis represents \(y^2\). Determine the values of the constants \(a\) and \(b\).

The law relating two variables is believed to be \(y = \frac{P}{x^2} + Q\).

(a) Explain what should be plotted on the axes to obtain a straight line and how \(P\) and \(Q\) can be found from this graph.
(b) The following table shows experimental values of \(x\) and \(y\):
x: 1, 2, 4, 5
y: 14.0, 6.5, 4.6, 4.4
Draw the linear graph and use it to estimate the values of \(P\) and \(Q\).
(c) A student suggests the law might actually be \(y = \frac{P}{x} + Q\). Explain how the plotted graph could be used to refute this suggestion.