Year 12 Maths Studies: Graphs of Cubic Functions Investigation

John McCarthyInvestigation of Cubic Functions

Year 12 Maths Studies: Graphs of Cubic Functions Investigation

John McCarthy

Introduction:
A cubic polynomial is any function of the form and exhibits both a concave and convex graph region. The following investigation aims to investigate the relationship between the two stationary points of a cubic polynomial and the point of inflection of that function. It will investigate situations where the leading coefficient of the function is equal, or not equal to , and situations where the expressions found do not apply. Finding this relationship will result in a simpler way of calculating the point of inflection of a cubic polynomial without having to find the second derivative.

Investigating Cubic Functions where the first coefficient is equal to :
The goal of this section is to investigate the nature of the two stationary points and the single point of inflection for cubic functions where the leading coefficient is equal to .Letting and equal the two stationary points and the point of inflection respectively, we can find these values for a set of sample functions.

For :

To find the two stationary points of this function we must find when .

Hence .
To find the point of inflection we must find when .

Substituting these values back into the original equation, we find the coordinates:

.

For :

Calculations are found in the Appendix.

For :

Calculations are found in the Appendix.

For :

Calculations are found in the Appendix.

This gives us:

Equation / A / B / G

From these values, it appears that the value of G is the midpoint of A and B. Given the general values A() and B(), we can then use the midpoint formula, which would suggest that the coordinates of the point of inflection are given by G

Proof of the conjecture for a leading coefficient of 1:
In this section, we will attempt to prove the conjecture that the point of inflection for a cubic with a leading coefficient of 1 is the midpoint of its two stationary points.

To find this, we must find the and values of the two stationary points and the and value of the point of inflection, and then test these with the midpoint formula.

Given a general cubic polynomial with a leading coefficient of : :

To find the values of the two stationary points, we must find when .

Hence the stationary points occur at and .

To find the value of the point of inflection we must find when .

Hence the point of inflection occurs at .

Substituting these values into the original equation, we can find the coordinates of the stationary points and the point of inflection.

For :

For :

For :

Using these coordinates for the two stationary points in the midpoint formula:

The result above is the coordinates of the point of inflection of the function, proving that the point of inflection is the midpoint of the two stationary points.

Investigating situations where this relationship does not apply:
In this section we will investigate situations where the values of the coefficients for the general formula of a cubic where the leading coefficient is equal to result in the above relationship not applying.

As is a constant in this equation that is removed when , and hence , are found, it has no limits on the value of the point of stationary points or the point of inflection. However, does affect the vertical translation of the cubic, as when values are substituted into the function, is always added on to the corresponding coordinates.

As the roots of negative numbers are imaginary, and hence undefined in real numbers, for the relationship to hold, . Hence .

If then the function no longer has any stationary points, as the expression for the coordinates of the stationary points becomes undefined. However, the expression for the point of inflection, , is unaffected, this suggests that in cubic polynomials where , there is a single point of inflection but no stationary points. In the event that , the stationary point of the function would occur at , in which case the point would be a stationary point of inflection.

An example of what the graphs would appear as for each of the above three cases:

Investigating Cubic Functions where the leading coefficient is not equal to :
The goal of this section is to investigate whether or not the relationship proposed above still holds for cubic polynomials where the leading coefficient is not equal to . Again, letting A, B and G be the two stationary points and the point of inflection respectively, we can find those points for the following samples:

For :

To find the stationary points we must find when .

Hence .

To find the point of inflection we must find when .

Substituting the values of into the original equation, we find the coordinates:

.

For :

Calculations are found in the Appendix.

For :

Calculations are found in the Appendix.

For :

Calculations are found in the Appendix.

This gives us:

Equation / A / B / G

From these values, it appears again that the point of inflection is the midpoint of the two stationary points, however this relationship does not hold true for the final equation, , as there are no stationary points for that function. We will investigate the reasons for this anomaly later.

Proof of the conjecture for a leading coefficient not equal to :
In this section we will attempt to prove the conjecture made above for cubic polynomials where the leading coefficient is not equal to .

To prove this relationship, we must find when the or values of the two stationary points and the or values of the point of inflection and then compare these with the midpoint of the two stationary points.

To attempt to prove this conjecture, we will use a general equation for a cubic polynomial:

To find the coordinates of the two stationary points, we must find when .

Hence the stationary points occur at and .

To find the coordinate of the point of inflection, we must find when .

Hence the point of inflection occurs at .

Substituting the values above into the general equation , we can attain the coordinates for each point.

For :

For :

For :

Using the coordinates of the two stationary points in the midpoint formula:

The resulting coordinates are the same as the coordinates of the point of inflection, proving that for all cubic polynomials of the form , the point of inflection exists as the midpoint of the two stationary points of the function.

Investigating situations where this relationship does not apply:
In this section we will investigate situations where the coefficients of the terms of the general function cause the above relationship to not hold. We have already observed one function where this does not hold true, found above to be . We will investigate if the following restrictions explain why this function did not have any stationary points.

As is a constant value that does not exist as a coefficient, and hence is removed in the calculations of the first and second derivatives of the cubic function, it has no effect on the values of the stationary points or the point of inflection. However does effect the vertical translation of the cubic polynomial, as it is added on to the corresponding value for any value of the function.

Given that the roots of negative numbers are imaginary, and hence undefined in real functions, , hence .

In the case that , then the expressions for the stationary points of the function become undefined, suggesting that the function has no stationary points, however the expression for the point of inflection for the function, remains unaffected, resulting in the functions till having a point of inflection even without stationary points. In the case that , the stationary points of the function would both come to be the same value at , in which case the function would have a single stationary point of inflection.

There is also a limitation on the value of , in that , as this would cause the point of inflection to be undefined. However, as this would effectively make the function a quadratic, for the purposes of this investigation the effects of this can be ignored.

Looking back at the equation found earlier that did not abide by the relationship found, , we see that in this equation, , and hence . In the above investigation, we found that when this is the case, the function has no stationary points, which is consistent with the results found for the stationary points of this function, proving that it does not invalidate the relationships found above.

An example of what a graph of a cubic would like for the above three cases:

Conclusion:

During this investigation, we have found the relationships between the two stationary points of a cubic polynomial and its point of inflection, in the case of the cubic polynomial having, or not having, a leading coefficient equal to . The relationship found is that the coordinates of the point of inflection are the midpoint of the two stationary points of the cubic function. We have also found the cases where this relationship does not apply, including when there is only a single stationary point that exists as a stationary point of inflection, and when there are no stationary points.

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