CURVE SKETCHING
Sarah Fox
THINGS YOU WILL BE FINDING WHEN
DOING CURVE SKETCHING
X-intercepts
 Vertical asymptotes
 Horizontal asymptotes
 First derivatives
 Sign lines
 Graphs of functions
 Graphing functions using properties of their
derivatives
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HOW TO FIND X-INTERCEPTS
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The purpose of finding x-intercepts is so that you
will know where your graph crosses the x-axis.
To do this you need to solve for x in the
numerator of the function you are going to graph.
Say your numerator is x2 – 4
You will then solve for x by setting x2 – 4 equal to
zero
So, your x-intercepts with be x= 2 and x= -2
HOW TO FIND VERTICAL ASYMPTOTES
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Vertical asymptotes are important to find
because they are vertical boundaries that your
graph cannot pass.
To find vertical asymptotes you have to solve for
x in the denominator in the same way that you
found x-intercepts:
Set the denominator equal to zero and solve for x.
HOW TO FIND HORIZONTAL ASYMPTOTES
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Horizontal asymptotes are important to find
because they are horizontal boundaries that your
graph can only pass through once.
The three rules on finding horizontal asymptotes
are:
If the power on bottom is greater then y= 0
 If the power on top is greater the asymptote is
oblique
 If the powers are equal use a ratio of the coefficients
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EXAMPLES OF HORIZONTAL ASYMPTOTES
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You’re given the function f(x) = _x2_
x2 + 3
This function would follow the third rule, the
ratio of coefficients because the powers in the
numerator and denominator are both squares, so
they are equal.
Since there are “invisible” ones in front of both of
the x2, the ratio would be 1 over 1 so the
horizontal asymptote would be y=1
EXAMPLES OF HORIZONTAL ASYMPTOTES 2
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Say you were given the function f(x) = _2x_
x2 - 1
This function would follow the first rule because
the power on bottom is greater so the horizontal
asymptote would be y=0
EXAMPLES OF HORIZONTAL ASYMPTOTES 3
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You are given the function: f(x) = _x2 - 1_
x
Since the power on top is greater, the horizontal
asymptote is oblique, and you will have to do long
division to find what y equals.
You will divide x2 – 1 by x, and you will find that
x goes into x2 – 1 “x” times with a remainder of -1
That will make your horizontal asymptote y=x
FIRST DERIVATIVES
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After finding your x-intercepts, vertical
asymptotes, and horizontal asymptotes, the next
step in curve sketching is taking the first
derivative of the function you are given.
The purpose of taking the first derivative is so
that you can solve for x in both the numerator
and denominator so you can make what is called
a sign line.
FIRST DERIVATIVES
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CONT.
Say you are given the function: f(x) = __x__
x2 + 1
You will use the quotient rule (bottom x
derivative of the top + top x derivative of the
bottom all over bottom squared) to find the first
derivative
After you use the quotient rule you should have:
f ’(x) = __(x2 + 1) (1) – (x) (2x)__
(x2 + 1)2
FIRST DERIVATIVE CONT.
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After you get: f ’(x) = __(x2 + 1) (1) – (x) (2x)__
(x2 + 1)2
You will have to simplify
You should then have: f ’(x) = __(1 - x) (1 + x)__
(x2 + 1)2
The things you would then use for your sign line
are: (1 – x), (1 + x), (x2 + 1), and (x2 + 1)
HOW TO MAKE A SIGN LINE
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Sign lines are important because they tell you the
integrals for which your graph is increasing or
decreasing from the first derivative
Using the last example: f ’(x) = __(1 - x) (1 + x)__
(x2 + 1)2
When making your sign line, things in the
numerator will use darkened circles, and things
from the denominator will used open circles
SIGN LINES CONT.
f ’(x) = __(1 - x) (1 + x)__
(x2 + 1)2
 (make sign line)
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GRAPHING
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Now you will have to put all of the stuff you
found (x-intercepts, vertical asymptotes,
horizontal asymptotes, and sign lines) together to
form your graph
Here’s an example: f(x) = __x__
2x – 1
First, you will need to find the x-intercepts,
which you should find to be
X=0
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GRAPHING CONT.
f(x) = __x__
2x - 1
 Then, you will need to find the vertical
asymptotes by solving for x in the denominator
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You should find it to be x = 1/2
Next you will find the horizontal asymptote by
going through the horizontal asymptote rules
You should find the horizontal asymptote to be
y = 1/2 since it’s a ratio of coifficients
GRAPHING CONT
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f(x) = __x__
2x - 1
Then you will have to take your first derivative
so you can make a sign line.
You should get your first derivative to be:
f ’(x) = __-1__
(2x – 1)2
GRAPHING CONT.
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f’(x) = __-1__
(2x – 1)2
Next, the things that will go on your sign line
should be: -1, 2x – 1, and 2x – 1
 (make sign line)
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GRAPHING CONT.
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f(x) = __x__
2x – 1
x- int: x = 0
v. a: x = 1/2
h. a: y = 1/2
TRY ME PROBLEM 1
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f(x) = _2x2 - 18_
x2-4
TRY ME PROBLEM 1 SOLUTION
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f ’(x) = __20x__
(x – 2)2 (x + 2)2
x- int: x = 3 x= -3
v. a: x = 2 x = -2
h. a: y = 2
TRY ME PROBLEM 2
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f(x) = __x__
(x2 + 2)1/2
TRY ME PROBLEM 2 SOLUTION
f ’(x) = __2__
(x2 + 2)3/2
x- int: x = 0
v. a: none
h. a: y = 1 y = -1
TRY ME PROBLEM 3
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f(x) = __x__
(x2 - 1)
TRY ME PROBLEM 3 SOLUTION
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f ’(x) = __- (1 + x2)__
(x - 1)2 (x + 1)2
x- int: x = 0
v. a: x = 1 x = -1
h. a: y = 0
HOW TO GRAPH FUNCTIONS USING PROPERTIES
OF THEIR DERIVATIVES
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Three rules to know when graphing functions
from their derivatives are:
If the graph of the first derivative is above the x-axis
the function is increasing. If it’s below the x-axis, the
function is decreasing.
 X-intercepts of the first derivative are max’s and
min’s of the function.
 Max’s and Min’s of the second derivative are
inflection points of the function. This shows where
concavity changes.
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TERMS TO KNOW
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Concavity: shows the shape of the graph.
Concave up “holds water”
 Concave down “spills water”
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A point of inflection, or inflection point, occurs
where the concavity changes direction
 A “max” is the highest point on the graph
 A “min” is the lowest point on the graph
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HOW TO…CONT.
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Example:
1985 AB 6 PART A
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Part A asks for where the relative max’s and
min’s are, and following rule # 2, you should
know that x-intercepts of the derivative are
max’s and min’s of the function.
And you see on the graph
that
that the x-intercepts are
x = -2 and x = 0
1985 AB 6 PART B
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Part B asks where on the graph is the function concave
up. We know from rule # 3 that:
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max’s and min’s of the second derivative are inflection points
of the function. This shows where concavity changes.
This is a good point to make a chart marking
each spot where the graph is concave up and
concave down.
 [ -3, -1) concave down
 ( -1, 1) concave up
 ( 1, 2) concave down
 ( 2, 3] concave up
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1985 AB 6 PART C
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Part C asks for a graph of the function
Now you have to use all of the information you
found out from the graph of the derivative like:
[-3, -2) increasing
 ( -2, 0) decreasing
 ( 0, 3] increasing
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[-3, -1) c.d.
( -1, 1) c.u.
( 1, 2) c.d.
( 2, 3] c.u.
1985 AB 6 PART C
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(graph function)
THE END
© Sarah Fox 2011