Two variables, \(x\) and \(y\), are related by an equation of the form \(y=A x^{b}\), where \(A\) and \(b\) are constants. The following pairs of values of \(x\) and \(y\) are given.

(b) Use your graph to find the values of \(A\) and \(b\).

Variables \(x\) and \(y\) are such that when \(\mathrm{e}^y\) is plotted against \(x^3\), a straight-line graph is obtained. This line passes through the points \((1,13.5)\) and \((7.5,0.5)\).

(a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which your equation is valid.

Variables \(x\) and \(y\) are such that when \(\ln y\) is plotted against \(x\), a straight-line graph is obtained. The line passes through the points \((1,\ln15)\) and \((2,\ln75)\).

Show that \(y=Ab^x\), where \(A\) and \(b\) are integers to be found.

When \(\ln y\) is plotted against \(x^3\), a straight line passing through the points \((2,5)\) and \((-8,25)\) is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find the value of \(x\) when \(y=\mathrm{e}^{25}\).

Variables \(x\) and \(y\) are such that when \(\sqrt{y}\) is plotted against \(x^{3}\) a straight line graph passing through the points \((2,5)\) and \((10,21)\) is obtained.

Find \(y\) in terms of \(x\).

When \(\mathrm e^y\) is plotted against \(x^2\), a straight-line graph with gradient \(-3\) is obtained.

The line passes through the point \((4.30,5.85)\).

(a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which \(y\) exists.

When \(\mathrm{e}^{5 y}\) is plotted against \(x^{3}\), a straight line passing through the points \((-2.56,4.38)\) and \((6.54,9.84)\) is obtained. (a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which \(y\) can exist.

The table shows the variables \(x\) and \(y\) which are related by the equation \(y=A b^{x^{2}}\), where \(A\) and \(b\) are constants.

Variables \(y\) and \(x\) are known to be connected by the relationship \(y=A b^{x}\) where \(A\) and \(b\) are constants. The table shows values of \(y\) for certain values of \(x\).

(b) Use your graph to find values of \(A\) and \(b\), giving each to 1 significant figure.

(c) Find an estimate of \(x\) when \(y=1500\).

When \(\mathrm{e}^{2 y}\) is plotted against \(x^{3}\), a straight line graph that passes through the points \((2,5)\) and \((6.4,7.2)\) is obtained. (a) Find \(y\) in terms of \(x\).

(b) Find the values of \(x\) for which \(y\) exists.

An experiment was carried out and values of \(y\) for certain values of \(x\) were recorded. The table shows the values recorded.

The relationship between \(y\) and \(x\) is modelled by \(y=A \mathrm{e}^{k x}, \quad\) where \(A\) and \(k\) are constants. (a) Draw a straight line graph for \(\ln y\) against \(x\).

(b) Find the equation of the line in part (a) and hence find the values of \(A\) and \(k\). Give each value correct to 1 significant figure.

(c) Find the value of \(x\) for which \(y=17\).

The table shows values of the variables \(x\) and \(y\), which are related by an equation of the form

\(y=Ab^{x^2},\)

where \(A\) and \(b\) are constants.

(a) Use the data to draw a straight line graph of \(\ln y\) against \(x^2\).

(b) Use your graph to estimate the values of \(A\) and \(b\). Give your answers correct to 1 significant figure.

(c) Estimate the value of \(y\) when \(x=1.75\).

(d) Estimate the positive value of \(x\) when \(y=20\).

The table shows values of the variables \(x\) and \(y\) which are related by an equation of the form \(y=Ax^b\), where \(A\) and \(b\) are constants.

(a) Use the data to draw a straight line graph of \(\ln y\) against \(\ln x\).

(b) Use your graph to estimate the values of \(A\) and \(b\).

(c) Estimate the value of \(x\) when \(y=100\).

Variables \(x\) and \(y\) are such that when \(\lg y\) is plotted against \(\sqrt{x}\), a straight line passing through the points \((1,5)\) and \((2.5,8)\) is obtained. Show that \(y=A\times b^{\sqrt{x}}\), where \(A\) and \(b\) are constants to be found.

Variables \(P\) and \(T\) are known to be connected by the relationship

\(P=Ab^T,\)

where \(A\) and \(b\) are constants. Values of \(P\) are found for certain values of time, \(T\).

(a) Show that a graph of \(\lg P\) against \(T\) will be a straight line.

(b) The diagram shows the graph of \(\lg P\) against \(T\). The graph passes through \((0,6)\) and \((14,12)\). Find the values of \(A\) and \(b\).

(c) Using the graph or otherwise, find the length of time for which \(P\) is between \(100\) million and \(1000\) million.

The table shows values of variables \(x\) and \(y\), which are related by the equation \(y=Ax^b\), where \(A\) and \(b\) are constants.

(a) Use the data to draw a straight line graph of \(\ln y\) against \(\ln x\).

(b) Use your graph to estimate the values of \(A\) and \(b\). Give your answers correct to 2 significant figures.

(c) Use your graph to estimate the value of \(y\) when \(x=3.5\).

When \(\ln(y+2)\) is plotted against \(x^2\), a straight line graph is obtained. The line passes through the points \((2.25,9.37)\) and \((4.75,3.92)\).

Find \(y\) in terms of \(x\).

Variables \(x\) and \(y\) are connected by the equation

\(y=Ax^b.\)

The straight line obtained by plotting \(\lg y\) against \(\lg x\) passes through the points \((0.61,0.57)\) and \((5.36,4.37)\).

(a) Find the value of \(A\) and of \(b\).

(b) Find the value of \(y\) when \(x=3\).

(c) Find the value of \(x\) when \(y=3\).

When \(e^y\) is plotted against \(x^2\), a straight line graph passing through the points \((2.24,5)\) and \((4.74,10)\) is obtained.

Find \(y\) in terms of \(x\).

Variables \(x\) and \(y\) are such that, when \(\ln y\) is plotted against \(\ln x\), a straight line graph passing through the points \((6,5)\) and \((8,9)\) is obtained.

Show that

\(y=e^{-7}x^2.\)

When \(e^{2y}\) is plotted against \(x^2\), a straight line graph passing through the points \((4,7.96)\) and \((2,3.76)\) is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find \(y\) when \(x=1\).

(c) Using your equation from part (a), find the positive values of \(x\) for which the straight line exists.

Variables \(x\) and \(y\) are such that when \(\sqrt y\) is plotted against \(\log_2(x+1)\), where \(x\gt -1\), a straight line is obtained which passes through \((2,10.4)\) and \((4,15.4)\).

(a) Find \(\sqrt y\) in terms of \(\log_2(x+1)\).

(b) Find the value of \(y\) when \(x=15\).

(c) Find the value of \(x\) when \(y=25\).

When \(\lg y\) is plotted against \(x^3\), a straight line is obtained. The line passes through the points \((6,7)\) and \((10,9)\), where the horizontal coordinate is \(x^3\) and the vertical coordinate is \(\lg y\).

Find \(y\) as a function of \(x\).

Variables \(x\) and \(y\) are such that, when \(\sqrt[4]{y}\) is plotted against \(\dfrac1x\), a straight line graph passing through the points \((0.5,9)\) and \((3,34)\) is obtained.

Find \(y\) as a function of \(x\).

Variables \(x\) and \(y\) are connected by the relationship

\(y=Ax^n,\)

where \(A\) and \(n\) are constants.

(a) Transform the relationship \(y=Ax^n\) to straight line form.

When \(\ln y\) is plotted against \(\ln x\), a straight line graph passing through the points \((0,0.5)\) and \((3.2,1.7)\) is obtained.

(b) Find the value of \(n\) and of \(A\).

(c) Find the value of \(y\) when \(x=11\).

The relationship between experimental values of two variables, \(x\) and \(y\), is given by \(y=Ab^x\), where \(A\) and \(b\) are constants.

(i) Transform the relationship \(y=Ab^x\) into straight line form.

The diagram shows \(\ln y\) plotted against \(x\) for ten different pairs of values of \(x\) and \(y\). The line of best fit has been drawn.

(ii) Find the equation of the line of best fit and the value, correct to 1 significant figure, of \(A\) and of \(b\).

(iii) Find the value, correct to 1 significant figure, of \(y\) when \(x=2.7\).

When \(\lg y\) is plotted against \(x^2\), a straight line graph is obtained which passes through the points \((2,4)\) and \((6,16)\).

(i) Show that \(y=10^{A+Bx^2}\), where \(A\) and \(B\) are constants.

(ii) Find \(y\) when \(x=\frac1{\sqrt3}\).

(iii) Find the positive value of \(x\) when \(y=2\).

When \(e^y\) is plotted against \(\frac1x\), a straight line graph passing through the points \((2,20)\) and \((4,8)\) is obtained.

(i) Find \(y\) in terms of \(x\).

(ii) Hence find the positive values of \(x\) for which \(y\) is defined.

(iii) Find the exact value of \(y\) when \(x=3\).

(iv) Find the exact value of \(x\) when \(y=2\).

When \(\lg y\) is plotted against \(x\), a straight line graph passing through the points \((2.2,3.6)\) and \((3.4,6)\) is obtained.

(i) Given that \(y=Ab^x\), find the value of each of the constants \(A\) and \(b\).

(ii) Find \(x\) when \(y=900\).

When \(\lg y^2\) is plotted against \(x\), a straight line is obtained passing through the points \((5,12)\) and \((3,20)\). Find \(y\) in terms of \(x\), giving your answer in the form \(y=10^{ax+b}\), where \(a\) and \(b\) are integers.

The table shows values of the variables \(x\) and \(y\) such that \(y=Ab^{x^2}\), where \(A\) and \(b\) are constants.

(i) Draw a straight line graph to show that \(y=Ab^{x^2}\).

(ii) Use your graph to find the value of \(A\) and of \(b\).

(iii) Estimate the value of \(x\) when \(y=100\).

Variables \(x\) and \(y\) are connected by the equation \(y=Ae^{bx}\), where \(A\) and \(b\) are constants. The table shows corresponding values of \(x\) and \(y\).

(i) Transform the equation \(y=Ae^{bx}\) into the equation of a straight line.

(ii) Use a suitable graph to show that \(x\) and \(y\) are connected by the equation \(y=Ae^{bx}\).

(iii) Use your graph to estimate the values of \(A\) and \(b\).

(iv) Estimate the value of \(x\) when \(y=500\).

(v) Estimate the value of \(y\) when \(x=5\).

The variables \(x\) and \(y\) are such that, when \(e^y\) is plotted against \(x^2\), a straight line graph passing through the points \((5,3)\) and \((3,1)\) is obtained. Find \(y\) in terms of \(x\).

An experiment was carried out recording values of \(y\) for certain values of \(x\). The variables \(x\) and \(y\) are thought to be connected by the relationship

\(y=ax^n,\)

where \(a\) and \(n\) are constants.

(i) Transform the relationship \(y=ax^n\) into straight line form.

The values of \(\ln y\) and \(\ln x\) were plotted and a line of best fit drawn.

(ii) Use the graph to find the value of \(a\) and of \(n\), stating the coordinates of the points that you use.

(iii) Find the value of \(x\) when \(y=50\).

An experiment was carried out recording values of \(y\) for certain values of \(x\). The variables \(x\) and \(y\) are thought to be connected by the relationship

\(y=ax^n,\)

where \(a\) and \(n\) are constants.

(i) Transform the relationship \(y=ax^n\) into straight line form.

The values of \(\ln y\) and \(\ln x\) were plotted and a line of best fit drawn.

(ii) Use the graph to find the value of \(a\) and of \(n\), stating the coordinates of the points that you use.

(iii) Find the value of \(x\) when \(y=50\).

The table shows values of the variables \(t\) and \(P\).

(i) Draw the graph of \(\ln P\) against \(t\) on the grid below.

(ii) Use the graph to estimate the value of \(P\) when \(t=2.2\).

(iii) Find the gradient of the graph and state the coordinates of the point where the graph meets the vertical axis.

(iv) Using your answers to part (iii), show that \(P=ab^t\), where \(a\) and \(b\) are constants to be found.

(v) Given that your equation in part (iv) is valid for values of \(t\) up to \(10\), find the smallest value of \(t\), correct to 1 decimal place, for which \(P\) is at least \(1000\).

Variables \(x\) and \(y\) are such that when \(\sqrt[3]{y}\) is plotted against \(\dfrac1x\), a straight line graph passing through the points \((0.2,5)\) and \((1,13)\) is obtained.

Express \(y\) in terms of \(x\).

When \(\lg y\) is plotted against \(x^2\), a straight line is obtained which passes through the points \((4,3)\) and \((12,7)\).

(i) Find the gradient of the line.

(ii) Use your answer to part (i) to express \(\lg y\) in terms of \(x\).

(iii) Hence express \(y\) in terms of \(x\), giving your answer in the form \(y=A(10^b)^{x^2}\), where \(A\) and \(b\) are constants.

When \(\lg y\) is plotted against \(x\), a straight line is obtained which passes through the points \((0.6,0.3)\) and \((1.1,0.2)\).

(i) Find \(\lg y\) in terms of \(x\).

(ii) Find \(y\) in terms of \(x\), giving your answer in the form \(y=A(10^b)^x\), where \(A\) and \(b\) are constants.

When \(\ln y\) is plotted against \(x^2\), a straight line is obtained which passes through the points \((0.2,2.4)\) and \((0.8,0.9)\).

(i) Express \(\ln y\) in terms of \(x^2\).

(ii) Hence express \(y\) in terms of \(z\), where \(z=e^{x^2}\).