WOMAN: In this interactive we explore polynomial graphs
of the form y = (x + 2) to the power of n (x – 1).
So we note from this general form
that these polynomials are always going to have two x-intercepts,
one at x = 1 and one at x = –2.
The x-intercept at x = 1 will be a point where the graph cuts the axis
due to the linear nature of this factor.
The behaviour at the x-intercept at x = –2, however, will vary,
depending on how we change the power of 'n'.
At the moment 'n' is set to one
so we're looking at the graph of y = (x + 2)(x – 1),
so that's a quadratic function
with x-intercepts at negative two and one.
Both factors are linear
and so we have the graph cutting the axis at both of these points.
If we increase the power of 'n'...
... the value of 'n' and hence the power of the first factor to two,
we'll now have a cubic function,
and we still see our x-intercepts at one and negative two.
But now we see we have a turning point
occurring at the x-intercept at x = –2,
and that's due to the squared power of that factor.
If we increase 'n' to three, we're now looking at a quartic function
and we're still seeing our two x-intercepts.
This time the cubed power on the (x + 2) factor
is causing a stationary point of inflection
at the x-intercept at x = –2.
Increasing the power again to four,
we've created a degree five polynomial here,
and we're again seeing a turning-point-like effect occurring
at x = –2
and still maintaining the graph cutting through the axis at x = 1.
Increasing the power again, we've now got a degree six polynomial,
and we're again looking at a point of inflect ...
... stationary point of inflection at x = –2.
Increasing the power again gives us a degree seven polynomial
with a stationary point at ... turning point,
stationary turning point at x = –2,
and still our x-intercept that cuts through the axis at x = 1.
And finally, making 'n' seven,
we now have a degree eight polynomial,
and, again, we're looking at a stationary point of inflection
occurring at x = –2
and a cut through the x-axis occurring at x = 1.
So we note that when we make 'n' an odd number,
and hence we have an odd power of (x + 2),
we create a point of inflection at that point,
or in the initial case, back when 'n' was equal to one,
we have a cut through the axis,
and when we have even powers of (x + 2)
we see a turning point occurring at that value of x = –2.
We also note the difference between the degree of the polynomial.
So when 'n' equals one
we have a degree two polynomial, or a quadratic,
and we note that these kinds of polynomials
tend to start up and finish up,
so they start in the same direction as they finish,
and we'll see that same thing occurring
for the degree four polynomial, for the degree six polynomial,
and for the degree eight polynomial.
Alternatively, for the even powers, even values of 'n'
and hence the odd-degree polynomials -
so in this case we have a cubic -
we see that the graph starts down at the left
and finishes up at the right.
So as 'x' gets smaller 'y' also gets more negative,
and as 'x' gets larger 'y' also gets larger and more positive,
and we see that occurring both for the cubic,
for the degree five polynomial, for the degree seven polynomial.
So there's many different patterns that we can observe
with polynomials of this form.