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Analyse a cubic graph or determine its rule

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Choosing a cubic representation

Different forms expose different features of a cubic. Factorised form displays roots and multiplicities, transformed form displays the stationary point of inflection, and general form is useful for four point conditions

\[P(x)=a\left(x-h\right)^3+k\]
  1. 1Match 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. 2Use 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. 3Keep multiplicity The exponent on a factor records how many times its root occurs \(\displaystyle x=-1\text{ has multiplicity }2\)
  4. 4Use the leading coefficient For a cubic, the sign of the leading coefficient determines the end behaviour \(\displaystyle a<0\quad\Rightarrow\quad y\to-\infty\text{ as }x\to\infty\)
Year 12, VCE Units 3 & 4, Chapter 6, 4 difficulty levels, No calculatorBuild a worksheet

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

\[P(x)=a\left(x-h\right)^3+k\]
  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

\[P(x)=a\left(x+2\right)^2\left(x-1\right),\quad P(0)=12\]
  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

Common mistakes

Losing a repeated root
✕\(\displaystyle 2\left(x+1\right)\left(x-3\right)\)✓\(\displaystyle 2\left(x+1\right)^2\left(x-3\right)\)
The exponent on a factor records the multiplicity of its root
Reversing a factor sign
✕\(\displaystyle x=-2\quad\Rightarrow\quad(x-2)\)✓\(\displaystyle x=-2\quad\Rightarrow\quad(x+2)\)
Subtract the root to form its factor; subtracting negative two gives the factor shown on the right
Reading end behaviour from the constant
✕\(\displaystyle P(0)>0\quad\Rightarrow\quad\text{the right end rises}\)✓\(\displaystyle a>0\quad\Rightarrow\quad\text{the left end falls and the right end rises}\)
The leading coefficient controls cubic end behaviour; the constant controls the vertical intercept

Difficulty levels

introChoose a cubic form from a stated root or stationary-point feature
standardConstruct a factorised or transformed rule from exact features and one additional condition
challengeUse multiplicities, translations or dilations to determine a cubic representation
examSelect a representation, construct the rule and interpret its roots or end behaviour

Questions about this topic

When is factorised form useful?+

It is useful when roots or root multiplicities are known because each factor displays a root and its exponent displays the multiplicity

What does a repeated root look like on a cubic graph?+

An even multiplicity usually makes the graph touch the horizontal axis at that root without crossing it. An odd multiplicity usually makes the graph cross

What controls the end behaviour of a cubic?+

The sign of the leading coefficient controls the directions of the two ends. The constant term does not control end behaviour

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 6: Applications of differentiation

In this chapter

Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter