Intersections of a cubic and a rule
Make the two outputs equal, solve the resulting polynomial equation for x and substitute each solution into an original rule for y.
\[P(x)=Q(x)\quad\Longleftrightarrow\quad P(x)-Q(x)=0\]
- Equate the outputs At an intersection, both rules have the same output for the same input \(\displaystyle P(x)=Q(x)\)
- Form a zero equation Move one rule to the other side so the difference is a polynomial equal to zero \(\displaystyle P(x)-Q(x)=0\)
- Solve for x The solutions of the difference equation are the possible horizontal coordinates \(\displaystyle x\left(x-2\right)\left(x+2\right)=0\quad\Rightarrow\quad x=-2,0,2\)
- Recover y Substitute each solution into either original rule to obtain the vertical coordinate \(\displaystyle Q(2)=3\quad\Rightarrow\quad (x,y)=(2,3)\)
Worked example
Find the intersections of these rules
\[P(x)=x^3-3x+2,\quad Q(x)=x+2\]
- \(\displaystyle P(x)=Q(x)\quad\Rightarrow\quad x^3-3x+2=x+2\)At an intersection, the two rules have equal outputs
- \(\displaystyle x^3-4x=0\quad\Rightarrow\quad x\left(x-2\right)\left(x+2\right)=0\quad\Rightarrow\quad x=-2,0,2\)Factoring the difference gives the three possible horizontal coordinates
- \(\displaystyle Q(-2)=0,\quad Q(0)=2,\quad Q(2)=4\)Substitution into the line gives the matching vertical coordinates
- \(\displaystyle \left(-2,0\right),\quad\left(0,2\right),\quad\left(2,4\right)\)Each intersection is written as an ordered pair of horizontal and vertical coordinates
Common mistakes
Stopping after solving for x
✕\(\displaystyle x=-2,0,2\)✓\(\displaystyle \left(-2,0\right),\quad\left(0,2\right),\quad\left(2,4\right)\)
An intersection is an ordered pair, so each horizontal coordinate needs its matching vertical coordinateSolving the rules separately
✕\(\displaystyle P(x)=0\quad\text{and}\quad Q(x)=0\)✓\(\displaystyle P(x)-Q(x)=0\)
An intersection compares the two outputs at the same input, so their difference must be zeroAdding the outputs
✕\(\displaystyle P(x)+Q(x)=0\)✓\(\displaystyle P(x)-Q(x)=0\)
Equal outputs give a zero difference, not a zero sumDifficulty levels
introFind the single real intersection of a cubic and a line or quadratic
standardSolve a factorisable cubic difference and state all intersection coordinates
challengeHandle repeated rational intersections or round irrational coordinates to the stated precision
examFind intersections and use the sign of P(x)-Q(x) to compare the two rules
Questions about this topic
Why are the two outputs equated?+
At an intersection, both rules produce the same y-value for the same x-value. Therefore P(x)-Q(x) is zero
Why must y be calculated after x?+
Solving the difference equation gives horizontal coordinates only. Substitution into an original rule supplies the corresponding vertical coordinates
How can the rules be compared between intersections?+
Evaluate the sign of P(x)-Q(x) in the interval. A positive sign means P is above Q there, and a negative sign means P is below Q.
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials
In this chapter
Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model