Analyse a parameterised polynomial family

Parameterised polynomial families

A parameterised polynomial uses the input variable for evaluation and a separate parameter to select a member. Substitute a given point to determine the parameter, then check any claimed family feature algebraically

1. Separate input and parameter The input variable is evaluated, while the parameter selects a member of the family \(\displaystyle x\text{ is the input},\qquad t\text{ selects a family member}\)
2. Substitute the given point Put the stated x-coordinate into the family rule and use the stated y-coordinate as the output \(\displaystyle P_t(3)=10\)
3. Solve for the parameter If the parameter coefficient is nonzero, solve the resulting exact equation. If it is zero, the point condition holds for every allowed parameter when y=A(x), and for none otherwise \(\displaystyle A(x)+tB(x)=y,\quad B(x)\ne0\Rightarrow t=\frac{y-A(x)}{B(x)}\)
4. Check a family claim A feature is shared only when the parameter term has the required value for every member \(\displaystyle B(h)=0\Rightarrow P_t(h)=A(h)\text{ for every }t\)

Worked example

Find k if the member of the family below passes through the point (0,0)

1. \(\displaystyle (0+2)^3=2^3=8\)Substitute the given x-coordinate into the parameter-free part of the rule
2. \(\displaystyle 0-4=-4\)Evaluate the factor multiplying the parameter at the same input
3. \(\displaystyle P_k(0)=8+k(-4)=0\Rightarrow8-4k=0\)Use the stated y-coordinate to form an equation in k
4. \(\displaystyle -4k=-8\Rightarrow k=2\)Divide both sides by −4 to recover the family parameter

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