Complete the table of values below for \(y=\dfrac12x^3-\dfrac12x^2-2x+2\).[1]
\(x\)
\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(3\)
\(y\)
\(-10\)
\(0\)
\(2\)
\(0\)
\(0\)
\(5\)
On the grid, draw the graph of \(y=\dfrac12x^3-\dfrac12x^2-2x+2\) for \(-3\le x\le3\).[3]
The water quality level of a pond over a 3-week period is modelled by the curve \(y=\dfrac12x^3-\dfrac12x^2-2x+2\), where \(x\) represents the time in weeks.
The minimum safe level for fish to survive is given by the line \(y=2-x\).
On the same grid, draw the line \(y=2-x\).[1]
From the graph, write down the value(s) of \(x\) where the water quality equals the minimum safe level.[2]
The fish are safe when the water quality is above the minimum safe level. Use your graph to find the range of values of \(x\) for which the fish are safe.[1]
The graph of \(y=ka^x\), where \(k\) and \(a\) are positive integers, passes through the points \((0,4)\) and \((3,32)\). Find the value of \(k\) and of \(a\).
\(k=\) [2]
\(a=\) [1]
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