7E - Families of cubic functions
Unlike quadratics, cubic functions do not have one standard shape when graphing. Instead, we need to consider several different cases based on the form of the equation:
Point of inflection form: \[P(x)=a(x-h)^3+k\]
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Linear factor form: \[P(x)=a(x-l)(x-m)(x-n)\]
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Repeated factor form: \[P(x)=a(x-m)^2(x-n)\]
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Irreducible factor form: \[P(x)=(x-m)(ax^2+bx+c)\]
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Notes on cubic graphs:
- All cubic graphs have at least one \(x\)-intercept (unless the domain is restricted).
- All cubic forms can be expanded to give the general form: \(y=ax^3+bx^2+cx+d\).
- Provided the domain is not restricted, the maximal domain of a cubic function is \(\mathbb{R}\).
- To find axial intercepts let \(x=0\) to find the \(y\)-intercept and let \(y=0\) to find the \(x\)-intercept(s).
Finding the turning points on a polynomial graph using CAS:
In Unit 2 we will use calculus to find the stationary points of different graphs. At the present, we will only find the turning points of a polynomial graph using a CAS calculator:
In Unit 2 we will use calculus to find the stationary points of different graphs. At the present, we will only find the turning points of a polynomial graph using a CAS calculator:
- Type the rule of the cubic function into the Main Menu of the CAS calculator.
- Highlight the rule and drag it into the graph screen.
- Use Analysis → G-Solve → Min/Max to find the decimal approximation of the turning points.
■ Warning! Unlike quadratics, the turning points of a cubic function do not exist halfway between the \(x\)-intercepts.
7E - Example 1: Finding the turning points of a polynomial graph (CAS)
Sketch the graph of \(y=x^3-2x^2-5x+6\) clearly labelling all axial intercepts and turning points with their coordinates. Where necessary, give the coordinates correct to two decimal places.
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7E - Example 1: Video solution
7E - Example 1: Practice
Question 1: ABC Question 2: ABC 7E - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Point of inflection form:
Cubic functions in the point of inflection form have the general equation \(y=a(x-h)^3+k\), where the point of inflection occurs at the point \((h,k)\). You may also note that this is also in transformation form, where:
Cubic functions in the point of inflection form have the general equation \(y=a(x-h)^3+k\), where the point of inflection occurs at the point \((h,k)\). You may also note that this is also in transformation form, where:
- \(a\) causes a dilation by a factor of \(a\) units from the \(x\)-axis. If \(a<0\) the graph is reflected over the \(x\)-axis.
- \(h\) causes a translation of \(h\) units parallel to the \(x\)-axis.
- \(k\) causes a translation of \(k\) units parallel to the \(y\)-axis.
Positive cubic graphs: \(a>0\)
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Negative cubic graphs: \(a<0\)
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7E - Example 2: Sketching cubic graphs in point of inflection form
Sketch the graph of \(y=(x-2)^3+1\) clearly labelling any axial intercepts and the point of inflection with their coordinates.
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7E - Example 2: Video solution
7E - Example 2: Practice
Question 1: ABC Question 2: ABC 7E - Example 2: Solutions
Question 1: ABC Question 2: ABC |
7E - Example 3: Sketching cubic graphs in point of inflection form
Sketch the graph of \(y=-(x+3)^3-27\) clearly labelling any axial intercepts and the point of inflection with their coordinates.
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7E - Example 3: Video solution
7E - Example 3: Practice
Question 1: ABC Question 2: ABC 7E - Example 3: Solutions
Question 1: ABC Question 2: ABC |
Linear factor form:
Some cubic functions can be expressed as a product of three linear factors \(y=a(x-l)(x-m)(x-n)\), where \((l,0)\), \((m,0)\) and \((n,0)\) are the \(x\)-intercepts of the graph.
Some cubic functions can be expressed as a product of three linear factors \(y=a(x-l)(x-m)(x-n)\), where \((l,0)\), \((m,0)\) and \((n,0)\) are the \(x\)-intercepts of the graph.
7E - Example 4: Sketching cubic graphs in linear factor form
Sketch the graph of \(y=\frac{1}{2}(x+4)(x-2)(x-5)\) on the set of axes provide. Clearly label any axial intercepts with their coordinates.
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7E - Example 4: Video solution
7E - Example 4: Practice
Question 1: ABC Question 2: ABC 7E - Example 4: Solutions
Question 1: ABC Question 2: ABC |
7E - Example 5: Sketching cubic graphs in linear factor form
Sketch the graph of \(y=-x^3+x^2+12x\) on the set of axes provide. Clearly label any axial intercepts with their coordinates.
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7E - Example 5: Video solution
7E - Example 5: Practice
Question 1: ABC Question 2: ABC 7E - Example 5: Solutions
Question 1: ABC Question 2: ABC |
Repeated factor form:
Some cubic functions can be factorised resulting in a repeated factor \(y=a(x-m)^2(x-n)\).
Some cubic functions can be factorised resulting in a repeated factor \(y=a(x-m)^2(x-n)\).
- The \(x\)-intercept at \((m,0)\) resembles a parabolic shape; that is, it is a turning point.
- The \(x\)-intercept at \((n,0)\) cuts the \(x\)-axis.
7E - Example 6: Sketching cubic graphs in repeated factor form
Sketch the graph of \(y=(x+2)^2(x-6)\) on the set of axes provide. Clearly label any axial intercepts with their coordinates.
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7E - Example 6: Video solution
7E - Example 6: Practice
Question 1: ABC Question 2: ABC 7E - Example 6: Solutions
Question 1: ABC Question 2: ABC |
Irreducible factor form:
Some cubic graphs will involve an irreducible quadratic factor \(y=(a-m)(ax^2+bx+c)\), where \(b^2-4ac<0\).
Some cubic graphs will involve an irreducible quadratic factor \(y=(a-m)(ax^2+bx+c)\), where \(b^2-4ac<0\).
- The graph will cut the \(x\)-axis at \((m,0)\).
- The graph will have no other \(x\)-axis intercepts; however, the graph may have turning points above or below the -axis. At this stage we will need to use a CAS calculator to graph these sorts of graphs.
7E - Example 7: Sketching cubic graphs with an IRREDUCIBLE quadratic factor (CAS)
Sketch the graph of \(y=-\frac{5}{3}(x+4)(x^2+2x+3)\) clearly labelling all axial intercepts and turning points with their coordinates. Where necessary, give the coordinates correct to two decimal places.
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7E - Example 7: Video solution
7E - Example 7: Practice
Question 1: ABC Question 2: ABC 7E - Example 7: Solutions
Question 1: ABC Question 2: ABC |