Modelling polynomials in grade 12

We have been looking at polynomial functions for the last few weeks.  We are getting comfortable with the equations in factored form, and understand how to draw graphs.  

Today we looked at tables of values, and the information held within them.

We looked for zeros, we noticed which of the finite differences is constant…

We drew graphs, and worked on making equations.

We determined that the constant 3rd difference played a part in determining the “a” value of the polynomial.

In fact, we determined that the constant differences in cubics, quartics, quintics etc all played a part in calculating the “a” value

In fact, the coefficient of “a” as shown above is equal to the factorial of the nth difference (the difference that is constant).  For a quartic, n! Would be 4! Which is 4x3x2x1=24

So, if we find that the 4th difference is constant, and the value is -12 we can determine the leading coefficient (the “a” value) by recognizing that it is a quartic, so 4!a=-12 which means 24a=-12 so a=-0.5

Our problem that we ended class with dealt with pyramidal numbers.

We want to determine f(n) and f(18).