Summary of graph drawing
Simple examples
linear functions Typically y = mx + b.
Point{slope formula y , b = m(x , a).
quadratic functions y = ax2 + bx + c.
Complete the square to get y = a(x , h)2 + k so vertex is (h; k).
common functions y = c; x; jxj; px; x2; x3 ; x1 .
Most of these graphs appear on rst page of.....
x1.2.
x2.1.
x1.5.
Six steps to drawing graphs
Given a rational function r(x) = f (x)=g(x) with integer coecients (this includes the case of a polynomial f (x)), draw y = r(x) as follows:
1. Is r(x) common function shifted/reected/stretched?
x1.5.
2. Plot y-intercepts.
3. (a) Plot x-intercepts, i.e., nd zeros of f (x) as follows:
i. does f (x) factor?
xP.3.
ii. Apply rational zero test
x2.5.
using Intermediate Value Theorem
x2.2.
then polynomial division
x2.3.
and quadratic formula,
xP.4.
remembering that you can't plot complex zeros!
x2.4.
(b) Plot vertical asymptotes, i.e., nd zeros of g(x).
x2.6.
4. Look at limiting behaviour as x ! 1:
(a) if it's a polynomial apply the leading coecient test.
x2.2.
(b) if it's a rational function nd the horizontal / slant asymptotes: x2.6.
i. if degree f < degree g then y = 0 is asymptote.
ii. if degree f = degree g then y = an =bn is asymptote, i.e., leading
coe of f divided by leading coe of g.
iii. if degree f > degree g then use long division to write
R(x) = q(x) + r(x)=g(x)
so that q(x) is a `slant' asymptote.
5. Use cunning and plot at least one point between and one point beyond all
zeros and vertical asymptotes.
6. Join dots with smooth curve.
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