Graphing Polynomial Functions
Key Questions
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A polynomial function in standard form must look like:
#f(x)=a_nx^n+a_(n-1)x^(n-1)+a_(n-2)x^(n-2)+...+a_2x^2+a_1x+a_0# where
#n in NN# and#a_i in ZZ, i in {0, 1, 2, ..., n}# If you are not familiar with the notation,
#NN# is the set of natural number, and#ZZ# is the set of integers.Examples of polynomial functions in standard form:
#f(x)=-4x^5+7x^3-6x^2+4#
#g(x)=3x^4-4x^3+6x-9# Examples of non polynomial functions:
#h(x)=4x^7-3x^5+sqrt(2x)-5#
#j(x)=-7x^3-2x^2+5/(x^3)#
#k(x)=4.5x^5-3x^2# I'll leave it to you to figure out why they are non polynomial functions.
Amory W.
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Sep 13 2014
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Answer:
This is quite a broad question.
Tips below.Explanation:
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.
Alan N. · 1 · Aug 12 2018