section 3-6
DESCRIPTION
Section 3-6. Curve Sketching. Steps to Analyze a Graph:. a) Intercepts and symmetry b) Domain and range (continuity) c) Asymptotes d) maximums and minimums e) Increasing & decreasing Points of inflection and Concavity graph. Intercepts. Intercepts: x-intercept: when y = 0 - PowerPoint PPT PresentationTRANSCRIPT
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SECTION 3-6Curve Sketching
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Steps to Analyze a Graph:
a) Intercepts and symmetry
b) Asymptotes
c) maximums and minimums
d) Increasing & decreasing
e) Points of inflection
f) Concavity
g) graph
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Intercepts• Intercepts:
x-intercept: when y = 0
y-intercept: when x =0
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SymmetryAbout the y-axis:
• Replace every x with –x if the function is
Symmetric about the y-axis
(all exponents are even)
About the origin:• Replace every x with –x if
the function is
symmetric about the origin
(all exponents are odd)
• About the x-axis: • not a function
)()( xfxf
)()( xfxf
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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
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Horizontal Asymptotes• BOBO BOTN EATS DC
• Bigger on bottom: y = 0• Bigger on top: none• Exponents are the same: divide coefficients
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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)
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Increasing and Decreasing
• Find critical points• First derivative test• Positive—increasing• Negative—decreasing increasing
increasing
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Inflection Points
Inflection points:
Set 2nd Derivative equal to zero and solve
Test for changes in concavity
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Concavity
2nd derivative test
Positive – concave up
Negative- concave down
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1) Sketch the curve which has the following:relative max
relative min
increasing on and
decreasing on
concave up
concave down
point of inflection
( ,0) ),2(
(0,2)
(1,)
( ,1)
(0,4)
(2,0)
(1,1)
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xxxf 63)( 2 2.) Sketch the graph of no calculator!
b) Asymptotes
a) Intercepts and symmetry
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xxxf 63)( 2 2.) Sketch the graph of
c) maximums and minimums
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xxxf 63)( 2 2.) Sketch the graph of
d) Increasing & decreasing
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xxxf 63)( 2 2.) Sketch the graph of
e) Points of inflection
f) Concavity
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xxxf 63)( 2 2.) Sketch the graph of
g) Graph
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25
2)(
2 x
xxf3.) Sketch the graph of
no calculator!
b) Asymptotes
a) Intercepts
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3.) Sketch the graph of 25
2)(
2 x
xxf
c) maximums and minimums
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3.) Sketch the graph of
d) Increasing & decreasing
25
2)(
2 x
xxf
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3.) Sketch the graph of 25
2)(
2 x
xxf
e) Inflection Points
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3.) Sketch the graph of 25
2)(
2 x
xxf
f) Concavity
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3.) Sketch the graph of
g) Graph
25
2)(
2 x
xxf