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WOMAN: In this interactive
we explore polynomial graphs

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of the form y = (x + 2)
to the power of n (x – 1).

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So we note from this general form

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that these polynomials are always
going to have two x-intercepts,

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one at x = 1 and one at x = –2.

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The x-intercept at x = 1 will be
a point where the graph cuts the axis

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due to the linear nature
of this factor.

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The behaviour at the x-intercept
at x = –2, however, will vary,

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depending on how we change
the power of 'n'.

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At the moment 'n' is set to one

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so we're looking at the graph
of y = (x + 2)(x – 1),

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so that's a quadratic function

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with x-intercepts
at negative two and one.

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Both factors are linear

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and so we have the graph cutting
the axis at both of these points.

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If we increase the power of 'n'...

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... the value of 'n' and hence the
power of the first factor to two,

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we'll now have a cubic function,

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and we still see our x-intercepts
at one and negative two.

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But now we see
we have a turning point

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occurring at the x-intercept
at x = –2,

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and that's due to the squared
power of that factor.

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If we increase 'n' to three, we're
now looking at a quartic function

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and we're still seeing
our two x-intercepts.

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This time the cubed power
on the (x + 2) factor

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is causing a stationary point
of inflection

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at the x-intercept at x = –2.

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Increasing the power again to four,

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we've created a degree five
polynomial here,

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and we're again seeing a
turning-point-like effect occurring

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at x = –2

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and still maintaining the graph
cutting through the axis at x = 1.

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Increasing the power again, we've now
got a degree six polynomial,

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and we're again looking
at a point of inflect ...

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... stationary point
of inflection at x = –2.

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Increasing the power again gives us
a degree seven polynomial

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with a stationary point
at ... turning point,

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stationary turning point at x = –2,

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and still our x-intercept that cuts
through the axis at x = 1.

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And finally, making 'n' seven,

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we now have a degree eight
polynomial,

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and, again, we're looking
at a stationary point of inflection

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occurring at x = –2

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and a cut through the x-axis
occurring at x = 1.

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So we note that when we make 'n'
an odd number,

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and hence we have
an odd power of (x + 2),

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we create a point of inflection
at that point,

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or in the initial case,
back when 'n' was equal to one,

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we have a cut through the axis,

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and when we have even powers
of (x + 2)

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we see a turning point occurring
at that value of x = –2.

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We also note the difference between
the degree of the polynomial.

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So when 'n' equals one

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we have a degree two polynomial,
or a quadratic,

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and we note that
these kinds of polynomials

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tend to start up and finish up,

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so they start in the same
direction as they finish,

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and we'll see that
same thing occurring

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for the degree four polynomial,
for the degree six polynomial,

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and for the degree eight polynomial.

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Alternatively, for the even powers,
even values of 'n'

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and hence the odd-degree
polynomials -

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so in this case we have a cubic -

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we see that the graph
starts down at the left

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and finishes up at the right.

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So as 'x' gets smaller
'y' also gets more negative,

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and as 'x' gets larger 'y' also
gets larger and more positive,

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and we see that occurring
both for the cubic,

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for the degree five polynomial,
for the degree seven polynomial.

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So there's many different patterns
that we can observe

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with polynomials of this form.