How do you graph a polynomial function?
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This is quite a broad question.
Tips below.
Let #f(x)# be a polynomial of #n^(th# degree with real coefficients.
To plot the graph of #f(x)# the following points are useful.
(i) Find the real zeros of #f(x)#, if any.
Set #f(x) =0# and solve for #x#.
The real zeros are points on the #x-#axis.
(ii) Find the #y-#intercept.
Find the point #f(0)#. This is the intercept on the #y-#axis.
(iii) Find the turning points of #f(x)#, if any.
Set #f'(x) = 0# and solve for #x#. (Say, #barx#)
Then,
where #f''(x_i)<0 -> f(x_i)# is a local maximum value.
where #f''(x_i)>0 -> f(x_i)# is a local minimum value.
where #f''(x_i)=0 -> f(x_i)# is an inflection point.
(iv) Plot points.
Outside of the above simply compute #f(x_j)# and plot points #(x_j, f(x_j))# as necessary to complete the graph.
I hope this helps.
