Logarithmic graph for power law

4
\includegraphics[max width=\textwidth, alt={}, center]{01f2de2b-3482-4694-889e-7fcd016b57e3-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

4
\includegraphics[max width=\textwidth, alt={}, center]{ad833f8c-80de-42ae-a186-93091a6fdf1e-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

2
\includegraphics[max width=\textwidth, alt={}, center]{9275a3ed-8820-481b-9fc8-28c21b81dbed-2_559_789_513_678} Two variable quantities \(x\) and \(y\) are related by the equation \(y = A x ^ { n }\), where \(A\) and \(n\) are constants. The diagram shows the result of plotting \(\ln y\) against \(\ln x\) for four pairs of values of \(x\) and \(y\). Use the diagram to estimate the values of \(A\) and \(n\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-2_456_725_1082_712} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(K\) and \(m\) correct to 2 significant figures.

3
\includegraphics[max width=\textwidth, alt={}, center]{9c26457d-4b65-4cd4-a9b9-128aba92dbf4-04_586_734_260_701} The variables \(x\) and \(y\) satisfy the equation \(y = k x ^ { a }\), where \(k\) and \(a\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(k\) and \(a\) correct to 3 significant figures.

4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants.
The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table below, showing values of \(X\), where \(X = x ^ { 2 }\).
2. On the diagram below, draw a linear graph relating \(X\) and \(y\).
3. Use your graph to find estimates, to two significant figures, for:
   1. the value of \(x\) when \(y = 15\);
   2. the values of \(a\) and \(b\).

4. \(\boldsymbol { x }\)	2	4	6	8
   \(\boldsymbol { X }\)
   \(\boldsymbol { y }\)	6.0	10.5	18.0	28.2

5. 
   \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-05_771_1586_1772_274}

7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants. Experimental evidence has provided the following approximate values:

1. On Figure 1, draw a linear graph connecting the variables \(X\) and \(Y\), where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
2. From your graph, find approximate values for the constants \(a\) and \(b\).