SECTION 3-6
Curve Sketching
Steps to Analyze a Graph:
a)
b)
c)
d)
e)
f)
g)
Intercepts and symmetry
Asymptotes
maximums and minimums
Increasing & decreasing
Points of inflection
Concavity
graph
Intercepts
• Intercepts:
x-intercept: when y = 0
y-intercept: when x =0
Symmetry
About the y-axis:
• Replace every x with –x if
f ( x)  f ( x) the function is
Symmetric about the y-axis
(all exponents are even)
About the origin:
• Replace every x with –x if
f ( x)   f ( x) the function is
symmetric about the origin
(all exponents are odd)
• About the x-axis:
• not a function
Asymptotes
• Only occur in rational functions
• Vertical: set denominator equal to zero
• Horizontal: take the limit as x approaches
infinity
• Slant: occur when the degree in the numerator
is one higher than the denominator
• Use long division
• Rewrite function as y = mx + b + remainder
• Remainder tends to zero as x approaches infinity, the
line y = mx + b is the asymptote
Horizontal Asymptotes
• BOBO BOTN EATS DC
• Bigger on bottom: y = 0
• Bigger on top: none
• Exponents are the same: divide coefficients
Maximums and Minimums
Use the first derivative test to find
the x values
Substitute x into the original
equation to obtain y-coordinate
Points: ordered pair (x,y)
Increasing and Decreasing
• Find critical points
• First derivative test
• Positive—increasing
• Negative—decreasing
increasing
increasing
Inflection Points
Inflection points:
Set 2nd Derivative equal
to zero and solve
Test for changes in
concavity
Concavity
2nd derivative test
Positive – concave up
Negative- concave down
1) Sketch the curve which has the following:
relative max (0,4)
relative min (2,0)
increasing on (,0) and (2, )

decreasing
on (0,2)
 up (1,)
concave
 down (,1)
concave
 inflection (1,1)
point of



2
f
(
x
)

3
x
 6x
2.) Sketch the graph of
no calculator!
a) Intercepts and symmetry
b) Asymptotes
2.) Sketch the graph of f ( x )  3 x 2  6 x
c) maximums and minimums
2.) Sketch the graph of f ( x )  3 x 2  6 x
d) Increasing & decreasing
2
f
(
x
)

3
x
 6x
2.) Sketch the graph of
e)
Points of inflection
f) Concavity
f (x)  3x 2  6 x
2.) Sketch the graph of
g) Graph
3.) Sketch the graph of
no calculator!
a) Intercepts
b) Asymptotes
2x
f ( x)  2
x  25
3.) Sketch the graph of
2x
f ( x)  2
x  25
c) maximums and minimums
3.) Sketch the graph of
2x
f ( x)  2
x  25
d) Increasing & decreasing
3.) Sketch the graph of
e) Inflection Points
2x
f ( x)  2
x  25
3.) Sketch the graph of
f) Concavity
2x
f ( x)  2
x  25
3.) Sketch the graph of
g) Graph
2x
f ( x)  2
x  25