MTH 173
Prof. Yan
Graph Sketch Guideline
The following checklist is intended as a guide to sketching a curve by hand. Not very item is relevant to every
function. (For instance, a given curve might not have an asymptote), but the guideline provide all information
you need to make a sketch that displays the most important aspects for the function.
a. Domain all 𝑥 where 𝑓(𝑥) is defined.
b. Intercepts
o The y–intercept is 𝑓(0) and this tells us where the curve intercept the y-axis.
o To find the x-intercepts, we set 𝑦 = 0 and solve for 𝑥. (you may use your calculator to find x-intercepts
if the equation is difficult to solve)
c. Symmetry
o If 𝑓 (𝑥) = 𝑓(−𝑥) for all x in the domain, then 𝑓 (𝑥) is an even function and the graph of is symmetric
about y-axis.
o If 𝑓 (𝑥) = −𝑓(−𝑥) for all x in the domain, then 𝑓 (𝑥) is an old function and the graph of is symmetric
about the origin.
d. Asymptotes
o Horizontal asymptotes. If either lim 𝑓(𝑥) = 𝐿 or lim 𝑓(𝑥) = 𝐿, then the line 𝑦 = 𝐿 is a horizontal
𝑥→∞
𝑥→−∞
asymptote of the curve 𝑦 = 𝑓(𝑥). If it turns out lim 𝑓(𝑥) = ∞(or −∞), then we don’ t have an
𝑥→∞
asymptote, but this is still helpful for sketching the curve.
o Vertical asymptotes. A line 𝑥 = 𝑐 is a vertical asymptote if
o lim+ 𝑓(𝑥) = ∞ (or −∞) or lim− 𝑓(𝑥) = ∞ (or −∞).
𝑥→𝑐
𝑥→𝑐
o Slant asymptotes (not required). If lim [𝑓(𝑥) − (𝑚𝑥 + 𝑏)] = 0, then the line 𝑦 = 𝑚𝑥 + 𝑏 is called a
𝑥→∞
slant asymptote of 𝑓(𝑥). (It occurs for a rational function when the degree of numerator is 1 larger than
the degree of detonator).
e. Interval of increasing or decreasing Compute 𝑓′(𝑥) and find the interval on which 𝑓′(𝑥) is positive (𝑓(𝑥)
is increasing) and the interval on which 𝑓′(𝑥) is negative (𝑓(𝑥) is decreasing).
f. Relative maximum and minimum Find the critical numbers of 𝑓(𝑥) (the numbers c where 𝑓 ′ (𝑐) = 0 or
𝑓′(𝑐) is undefined). If 𝑓’(𝑥) changes from positive to negative at a critical number 𝑐, then (𝑐, 𝑓 (𝑐)) is a
relative maximum. If 𝑓’(𝑥) changes from negative to positive at a critical number 𝑐, then (𝑐, 𝑓 (𝑐)) is a
relative minimum. Otherwise, (𝑐, 𝑓 (𝑐)) is neither.
g. Concavity and inflection points Compute 𝑓′′(𝑥) and use Concavity test. The curve of 𝑓 (𝑥) is concave up
′
where 𝑓 ′′ (𝑥) > 0 and concave down where 𝑓 ′ (𝑥) < 0. The points of inflection is where the direction of
concavity changes.
h. Sketch the graph Use the above information to sketch the graph. Sketch the asymptote as dashed lines.
Plot intercepts, all relative maximum and minimum points, and points of inflection. Then make the curve
pass through these points, falling and rising according to E, with concativity according to G, and approaching
the asymptotes.
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