Intermediate Mathematics - Graphs of polynomial functions
Graphs of polynomial functions
You are required to be able to draw graphs of polynomial functions y=f(x),
up to degree 4 in factorised form that have linear factors. To draw the
graph of a polynomial function:
* express it in factorised form
* use your knowledge to imagine the basic shape of the graph
* look for repeated factors. If the power of the factor is even, eg. (x -
c) or (x - c), the graph will touch the X-axis and have a turning point at
(c, 0). If the repeated factor is (x - c), the graph will have a stationary
point of inflection at (c, 0).
* find the x intercepts: set f(x) = 0 with f(x) in factorised form
* find the y intercept: set x = 0 . The constant term in f(x), written
out in expanded form, gives the y intercept. Expansion is not necessary as
f(0) can often be calculated easily from the factorised form.
* identify turning points. For now, where possible, the coordinates will
be found algebraically. If this is difficult, simply identify the number of
turning points from your knowledge of the shape of the graph. See the
Calculus section for an alternative method.
* determine the sign of f(x) for different regions of the domain
* sketch the graph
* check with graphics calculator.
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