Search topics

Search

Analyse a parameterised polynomial family

Start a set of 8
Step 2 of 5

Analyse a parameterised family

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

\[P_t(x)=A(x)+tB(x)\]
  1. 1Separate 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. 2Substitute 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. 3Solve 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. 4Check a family claim A feature is shared only when the parameter term has the required value for every member \(\displaystyle B(h)=0\Rightarrow P_t(h)=A(h)\text{ for every }t\)
Year 11, VCE Units 1 & 2, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

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

\[P_t(x)=A(x)+tB(x)\]
  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)

\[P_k(x)=(x+2)^3+k(x-4),\qquad P_k(0)=0\]
  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

Common mistakes

Using the parameter as the input
✕\(\displaystyle P_t(x)=(x-1)^3+t(x-1),\quad(3,10),\quad x=t=3\Rightarrow P_3(3)=8+6=14\)✓\(\displaystyle P_t(x)=(x-1)^3+t(x-1),\quad(3,10),\quad x=3,\qquad P_t(3)=8+2t=10\)
The parameter selects a member, while x remains the input in the point condition
Generalising one member
✕\(\displaystyle P_t(x)=(x-2)^3+t(x+1),\quad P_1(0)=-7\Rightarrow P_t(0)=-7\)✓\(\displaystyle P_t(x)=(x-2)^3+t(x+1),\quad P_t(0)=-8+t\)
A value from one member cannot be assigned to every member without checking the parameter term
Omitting the parameter term
✕\(\displaystyle P_t(x)=(x+2)^3\)✓\(\displaystyle P_t(x)=(x+2)^3+t(x-4)\)
The parameter term is the part of the rule that changes between family members and must be retained

Difficulty levels

introA visible feature controlled by one parameter is identified for the family
standardOne exact point determines a parameter before member features are recorded
challengePositive and negative cases or a fixed feature are compared using coefficient data
examInvariant analysis, parameter recovery, graph choice and a check of the supplied point are combined

Questions about this topic

What is the difference between an input and a family parameter?+

The input is substituted into a selected polynomial rule. The parameter selects which rule, or member, from the family is being used

How can a point determine a family parameter?+

Substitute the point's coordinates into the family rule. A unique parameter follows when its coefficient at that input is nonzero. If the coefficient is zero, the point lies on every allowed member when its output equals the parameter-free part, and on none otherwise

When is a feature shared by every member of a family?+

The parameter-dependent part must have the required common value, often zero, at the input or coefficient being considered

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 5: Higher-degree polynomials

In this chapter

Interpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityAnalyse a parameterised polynomial familyBracket and approximate a polynomial root by bisectionApproximate solutions from intersections of two graphsLocate a polynomial turning point numerically