Analyse a cubic graph or determine its rule

Representations of a cubic

Choose the form that shows the information supplied. Factorised form shows roots, transformed form shows a stationary inflection, and general form supports coefficient conditions

1. Match form to information Use transformed-cube form when a stationary point of inflection is the key feature \(\displaystyle P(x)=a\left(x-h\right)^3+k\)
2. Use factors for roots Use factorised form when roots or their multiplicities are supplied \(\displaystyle P(x)=2\left(x+1\right)^2\left(x-3\right)\)
3. Keep multiplicity The exponent on a factor records how many times its root occurs \(\displaystyle x=-1\text{ has multiplicity }2\)
4. Use the leading coefficient For a cubic, the sign of the leading coefficient determines the end behaviour \(\displaystyle a<0\quad\Rightarrow\quad y\to-\infty\text{ as }x\to\infty\)

Worked example

A cubic has a double root at negative two, a simple root at one, and passes through the point (0, 12). Determine its rule and check the output at two

1. \(\displaystyle P(x)=a\left(x+2\right)^2\left(x-1\right)\)The repeated root gives the squared factor and the other root gives the linear factor
2. \(\displaystyle P(0)=a\left(2\right)^2\left(-1\right)=-4a=12\)Substitute the known point into the factorised rule
3. \(\displaystyle a=-3\quad\Rightarrow\quad P(x)=-3\left(x+2\right)^2\left(x-1\right)\)Solving for the scale factor gives negative three
4. \(\displaystyle P(2)=-3\left(4\right)^2\left(1\right)=-48\)Substitution checks the completed rule at two

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