CHAPTER 3 SECTION 3.6 CURVE SKETCHING

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CHAPTER 3SECTION 3.6CURVE SKETCHING
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Graph the following functions on your calculator
and make a sketch.What do you notice about the
graph of a function where there is a factor that
appears twice?

• 1) y(x1)(x-2)2 2) y x(x3)2 3)
  y(x1)(x-3)2

Where there is a factor that appears twice
(double root) the graph bounces at that zero.
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Graph the following functions on your calculator
and make a sketch.What do you notice about the
graph of a function where there is a factor that
appears three times?

• 1) yx3(x-3) 2) y (x2)(x-1)3 3)
  y(x-2)(x1)3

Where there is a factor that appears three
times (triple root) the graph wiggles at that
zero.
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Graph the following functions on your calculator
and make a sketch.What do you notice about the
graph of a function where there is a factor that
appears four times?

• 1) yx4(x-2) 2) y (x-1)4(x1) 3)
  y(x1)4(x-1)

Where there is a factor that appears four times
the graph bounces at that zero.
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What do you think the graph of a function would
look like at the zero that corresponds to a
factor that appears n times?

• If a linear factor appears once the graph goes
  through the x-axis at that zero
• If a linear factor appears an even number of
  times- the graph bounces at that zero.
• If a linear factor appears an odd number of times
  (greater than 1) the graph wiggles at that
  zero.

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What will the graph look like?

• f(x) x(x-5)3(x4)

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What will the graph look like?

• y (x-1)2 (x3)(x1)5

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Write an equation for the graph.
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Write an equation for the graph.
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Write an equation for the graph.
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DISAIMIS
OMAIN
NTERCEPTS
YMMETRY
SYMPTOTES
NTERVALS
AX MIN
NFLECTION
KETCH
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Strategy

• USE ALGEBRA FIRST WITH A T-CHART
• Determine domain of function
• Find y-intercepts, x-intercepts (zeros)
• Check for vertical, horizontal asymptotes
• Determine values for f '(x) 0, critical points
• Determine f ''(x)
• Gives inflection points
• Test for intervals of concave up, down
• Plot intercepts, critical points, inflection
  points
• Connect points with smooth curve
• Check sketch with graphing calculator

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GUIDELINES FOR SKETCHING A CURVE

• The following checklist is intended as a guide to
  sketching a curve y f(x) by hand.
• Not every item is relevant to every function.
• For instance, a given curve might not have an
  asymptote or possess symmetry.
• However, the guidelines provide all the
  information you need to make a sketch that
  displays the most important aspects of the
  function.

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A. DOMAIN

• Its often useful to start by determining the
  domain D of f.
• This is the set of values of x for which f(x) is
  defined.

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B. INTERCEPTS

• The y-intercept is f(0) and this tells us where
  the curve intersects the y-axis.
• To find the x-intercepts, we set y 0 and solve
  for x.
• You can omit this step if the equation is
  difficult to solve.

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C. SYMMETRYEVEN FUNCTION

• If f(-x) f(x) for all x in D, that is, the
  equation of the curve is unchanged when x is
  replaced by -x, then f is an even function and
  the curve is symmetric about the y-axis.
• This means that our work is cut in half.

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C. SYMMETRYEVEN FUNCTION

• If we know what the curve looks like for x 0,
  then we need only reflect about the y-axis to
  obtain the complete curve.

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C. SYMMETRYEVEN FUNCTION

• Here are some examples
• y x2
• y x4
• y x
• y cos x

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C. SYMMETRYODD FUNCTION

• If f(-x) -f(x) for all x in D, then f is an
  odd function and the curve is symmetric about the
  origin.

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C. SYMMETRYODD FUNCTION

• Again, we can obtain the complete curve if we
  know what it looks like for x 0.
• Rotate 180
• about the origin.

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C. SYMMETRYODD FUNCTION

• Some simple examples of odd functions are
• y x
• y x3
• y x5
• y sin x

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C. SYMMETRYPERIODIC FUNCTION

• If f(x p) f(x) for all x in D, where p is a
  positive constant, then f is called a periodic
  function.
• The smallest such number p is called the period.
• For instance, y sin x has period 2p and y tan
  x has period p.

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C. SYMMETRYPERIODIC FUNCTION

• If we know what the graph looks like in an
  intervalof length p, then we can use translation
  to sketchthe entire graph.

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D. ASYMPTOTESHORIZONTAL

• Recall from Section 2.6 that, if either
  or , then the line y
  L is a horizontal asymptote of the curve y f
  (x).
• If it turns out that (or -8),
  then we do not have an asymptote to the right.
• Nevertheless, that is still useful information
  for sketching the curve.

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D. ASYMPTOTESVERTICAL
Equation 1

• Recall that the line x a is a vertical
  asymptote if at least one of the following
  statements is true

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D. ASYMPTOTESVERTICAL

• For rational functions, you can locate the
  vertical asymptotes by equating the denominator
  to 0 after canceling any common factors.
• However, for other functions, this method does
  not apply.

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D. ASYMPTOTESVERTICAL

• Furthermore, in sketching the curve, it is very
  useful to know exactly which of the statements in
  Equation 1 is true.
• If f(a) is not defined but a is an endpoint of
  the domain of f, then you should compute or
  , whether or not this limit is infinite.

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D. ASYMPTOTESSLANT

• Slant asymptotes are discussed at the end of this
  section.

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E. INTERVALS OF INCREASE OR DECREASE

• Use the I /D Test.
• Compute f(x) and find the intervals on which
• f(x) is positive (f is increasing).
• f(x) is negative (f is decreasing).

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F. LOCAL MAXIMUM AND MINIMUM VALUES

• Find the critical numbers of f (the numbers c
  where f(c) 0 or f(c) does not exist).
• Then, use the First Derivative Test.
• If f changes from positive to negative at a
  critical number c, then f(c) is a local maximum.
• If f changes from negative to positive at c,
  then f(c) is a local minimum.

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F. LOCAL MAXIMUM AND MINIMUM VALUES

• Although it is usually preferable to use the
  First Derivative Test, you can use the Second
  Derivative Test if f(c) 0 and f(c) ? 0.
• Then,
• f(c) gt 0 implies that f(c) is a local minimum.
• f(c) lt 0 implies that f(c) is a local maximum.

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G. CONCAVITY AND POINTS OF INFLECTION

• Compute f(x) and use the Concavity Test.
• The curve is
• Concave upward where f(x) gt 0
• Concave downward where f(x) lt 0

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G. CONCAVITY AND POINTS OF INFLECTION

• Inflection points occur where the direction of
  concavity changes.

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H. SKETCH AND CURVE

• Using the information in items AG, draw the
  graph.
• Sketch the asymptotes as dashed lines.
• Plot the intercepts, maximum and minimum points,
  and inflection points.
• Then, make the curve pass through these points,
  rising and falling according to E, with
  concavity according to G, and approaching the
  asymptotes

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H. SKETCH AND CURVE

• If additional accuracy is desired near any
  point, you can compute the value of the
  derivative there.
• The tangent indicates the direction in which the
  curve proceeds.

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Example 1

• Use the guidelines to sketch the curve

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Example 1

• A. The domain is x x2 1 ? 0 x
  x ? 1 (-8, -1) U (-1, -1) U (1, 8)
• B. The x- and y-intercepts are both 0.

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Example 1

• C. Since f(-x) f(x), the function is even.
• The curve is symmetric about the y-axis.

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Example 1

• D. Therefore, the line y 2 is a horizontal
  asymptote.

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Example 1

• Since the denominator is 0 when x 1, we
  compute the following limits
• Thus, the lines x 1 and x -1 are vertical
  asymptotes.

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Example 1

• This information about limits and asymptotes
  enables us to draw the preliminary sketch,
  showing the parts of the curve near the
  asymptotes.

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Example 1
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Example 1

• F. The only critical number is x 0.
• Since f changes from positive to negative at 0,
  f(0) 0 is a local maximum by the First
  Derivative Test. (1 and -1 are not in the
  domain!!!!!!)

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Example 1
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Example 1

• It has no point of inflection since 1 and -1 are
  not in the domain of f.!!!!!!!!!!!

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Example 1

• H. Using the information in EG, we finish the
  sketch.

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Example 2

• Sketch the graph of

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Example 2

• A. Domain x x 1 gt 0 x x gt -1
  (-1, 8)
• B. The x- and y-intercepts are both 0.
• C. Symmetry None

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Example 2

• D. Since , there is no
  horizontal asymptote.
• Since as x ? -1 and f(x) is
• always positive, we have
  , and so the line x -1 is a vertical asymptote

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Example 2

• E.
• We see that f(x) 0 when x 0 (notice that
  -4/3 is not in the domain of f).
• So, the only critical number is 0.

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Example 2

• As f(x) lt 0 when -1 lt x lt 0 and f(x) gt 0 when x
  gt 0, f is
• Decreasing on (-1, 0)
• Increasing on (0, 8)

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Example 2

• F. Since f(0) 0 and f changes from negative
  to positive at 0, f(0) 0 is a local (and
  absolute) minimum by the First Derivative Test.

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Example 2

• G.
• Note that the denominator is always positive.
• The numerator is the quadratic 3x2 8x 8,
  which is always positive because its
  discriminant is b2 - 4ac -32, which is
  negative, and the coefficient of x2 is positive.

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Example 2

• So, f(x) gt 0 for all x in the domain of f.
• This means that
• f is concave upward on (-1, 8).
• There is no point of inflection.

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Example 2

• H. The curve is sketched here.

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• Sketch the graph of

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Example 3

• A. The domain is R
• B. The y-intercept is f(0) ½. The x-intercepts
  occur when cos x 0, that is, x (2n 1)p/2,
  where n is an integer.

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Example 3

• C. f is neither even nor odd.
• However, f(x 2p) f(x) for all x.
• Thus, f is periodic and has period 2p.
• So, in what follows, we need to consider only 0
  x 2p and then extend the curve by translation
  in part H.
• D. Asymptotes None

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Example 3

• E.
• Thus, f(x) gt 0 when 2 sin x 1 lt 0
  sin x lt -½ 7p/6 lt x lt 11p/6

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Example 3

• Thus, f is
• Increasing on (7p/6, 11p/6)
• Decreasing on (0, 7p/6) and (11p/6, 2p)

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Example 3

• F. From part E and the First Derivative Test, we
  see that
• The local minimum value is f(7p/6) -1/
• The local maximum value is f(11p/6) -1/

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Example 3

• G. If we use the Quotient Rule again and
  simplify, we get
• (2 sin x)3 gt 0 and 1 sin x 0 for all x.
• So, we know that f(x) gt 0 when cos x lt 0, that
  is, p/2 lt x lt 3p/2.

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Example 3

• Thus, f is concave upward on (p/2, 3p/2) and
  concave downward on (0, p/2) and (3p/2, 2p).
• The inflection points are (p/2, 0) and (3p/2, 0).

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Example 3

• H. The graph of the function restricted to 0 x
  2p is shown here.

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Example 3

• Then, we extend it, using periodicity, to the
  complete graph here.

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SLANT ASYMPTOTES

• Some curves have asymptotes that are
  obliquethat is, neither horizontal nor vertical.

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SLANT ASYMPTOTES

• For rational functions, slant asymptotes occur
  when the degree of the numerator is one more than
  the degree of the denominator.
• In such a case, the equation of the slant
  asymptote can be found by long divisionas in
  following example.

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Example 6

• Sketch the graph of

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SLANT ASYMPTOTES
Example 6

• A. The domain is R (-8, 8)
• B. The x- and y-intercepts are both 0.
• C. As f(-x) -f(x), f is odd and its graph is
  symmetric about the origin.

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SLANT ASYMPTOTES
Example 6

• Since x2 1 is never 0, there is no vertical
  asymptote.
• Since f(x) ? 8 as x ? 8 and f(x) ? -8 as x ? -
  8, there is no horizontal asymptote.

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SLANT ASYMPTOTES
Example 6

• However, long division gives
• So, the line y x is a slant asymptote.

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SLANT ASYMPTOTES
Example 6

• E.
• Since f(x) gt 0 for all x (except 0), f is
  increasing on (- 8, 8).

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SLANT ASYMPTOTES
Example 6

• F. Although f(0) 0, f does not change sign at
  0.
• So, there is no local maximum or minimum.

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SLANT ASYMPTOTES
Example 6

• G.
• Since f(x) 0 when x 0 or x , we
  set up the following chart.

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SLANT ASYMPTOTES
Example 6

• The points of inflection are
• (0, 0)

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SLANT ASYMPTOTES
Example 6

• H. The graph of f is sketched.

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First derivative
Curve is rising.
Curve is falling.
Possible local maximum or minimum.
Second derivative
Curve is concave up.
Curve is concave down.
Possible inflection point (where concavity
changes).
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Example
Graph
We can use a chart to organize our thoughts.
First derivative test
negative
positive
positive
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Example
Graph
First derivative test
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Example
Graph
NOTE On the AP Exam, it is not sufficient to
simply draw the chart and write the answer. You
must give a written explanation!
First derivative test
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Example
Graph
Or you could use the second derivative test
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Example
Graph
We then look for inflection points by setting the
second derivative equal to zero.
negative
positive
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Make a summary table
p
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• Sketch

• Include intercepts, asymptotes, extrema,
  inflection points, and intervals of concavity.

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• Sketch

VA at x 2

• Include intercepts, asymptotes, extrema,
  inflection points, and intervals of concavity.

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Intervals
Intercepts
Test values
f (test pt)
f(x)
f (test pt)
f(x)
Asymptotes
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Intervals
Intercepts
Test values
f (test pt)
No x-int
f(x)
f (test pt)
f(x)
rel min
No inf pts
rel max
Asymptotes
Oblique Asymptote at
remainder
VA at x 2
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Example
Sketch
1. Domain (-8, 8).
2. Intercept (0, 1)
3.

1. No Asymptotes

5.
f inc. on (-8, 1) U (3, 8), dec. on (1, 3).
6. Relative max. (1, 5) relative min. (3,
1)
7.
f concave down (-8, 2) up on (2, 8).
8. Inflection point (2, 3)
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Sketch
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Example
Sketch
1. Domain x ? -3
2. Intercepts (0, -1) and (3/2, 0)
3.

1. Horizontal y 2 Vertical x -3

5.
f is increasing on (-8,-3) U (-3, 8).
6. No relative extrema.
f is concave up on (-8,-3) and f is concave
down on (-3, 8).
7.
8. No inflection points
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Sketch
y 2
x -3
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• Sketch

• Include intercepts, asymptotes, extrema,
  inflection points, and intervals of concavity.

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• Sketch

• Include intercepts, asymptotes, extrema,
  inflection points, and intervals of concavity.

Whats the difference between f(x) and g(x)?
Compare to
The abs val makes all the negative parts of g(x)
reflect above the x-axis.
Rel min at (3, -4)
Parabola opening upward (no asymptotes)
Always concave up
x-int
y-int
g(x)
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x-int x 1 and 5
y-int y 5
f(x)
Rel max at (3, 4)
Rel min at (1, 0)
Rel min at (5, 0)
Inf pts none