2. What should you consider?
•
•
•
•
•
•
•
•
•
symmetries
x-intercept
y-intercept
relative extrema
asymptores
concavity
inflection points
intervals of increase
interval of decrease
ASYMPTOTE – VA HA SA
N(x) ax m
Let R(x) =
=
D(x) bx n
VERTICAL ASYMPTOTE
If
D(x) = 0, then VA : x = c
HORIZONTAL ASYMPTOTE
If
m > n,
then HA : doesn't exist
If
m < n,
then HA : y = 0,
If
m = n,
then HA : y =
SLANT ASYMPTOTE
If
x - axis
a
b
m = n +1, then SA : y = mx + b
Long divide N(x) by D(x)
3. Let’s see how that
works!!!
2x 2 - 8
Sketch the graph of the equation f (x) = 2
x -16
2x - 8
=0
2
x -16
2
VA:
2x 2 - 8 = 0
x 2 -16 = 0
f '(x) =
f '(x) =
2 × 02 - 8 1
f (0) = 2
=
0 -16 2
x = -2, 2
x = -4, 4
HA:
4x ( x 2 -16) - 2x ( 2x 2 - 8)
(x
(x
2
-16)
16 ( x - 4)
2
-16) ( x 2 -16)
-
f '(x) =
2
=
16 ( x - 4)
·
-4
·
4
4x 3 - 64 - 4x 3 +16x
(x
2
-16)
f '(x) =
( x + 4) ( x - 4) ( x 2 -16)
-
y=2
m=n
+
2
=
16x - 64
(x
2
-16)
2
16
=0
2
( x + 4) ( x -16)
4. Almost Done !!!!
f "(x) =
-16 é( x 2 -16) + 2x ( x + 4)ù
ë
û
é( x + 4) ( x -16)ù
ë
û
2
2
-16 é x -16 + 2x + 8xù
ë
û
2
f "(x) =
f "(x) =
=0
2
é( x + 4) ( x 2 -16)ù
ë
û
2
-16 (3x - 4) ( x + 4)
( x + 4 ) ( x + 4) ( x
-
2
-16)
·
-4
2
=0
=0
f "(x) =
-16 é3x 2 + 8x -16ù
ë
û
é( x + 4) ( x -16)ù
ë
û
f "(x) =
2
2
-16 (3x - 4)
( x + 4) ( x
- · + · 4/3
4
=0
2
-16)
2
=0
5. Let’ make a list
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•
•
•
•
•
•
•
•
-2, 2
x-intercept
0.5
y-intercept
-4, 4
VA
2
HA
________
SA
[ 4, +¥)
f(x)
( -¥, 4]
f(x)
______
X max
X min
4
• f(x) È
• f(x) Ç
• X infl
æ4 ö
ç , 4÷
è3 ø
æ
4ö
-¥, ÷ and ( 4, +¥)
ç
è
3ø
4
3
6. Sketch the graph of the equation
•
•
•
•
•
•
•
•
•
x-intercept
y-intercept
VA
HA
SA
f(x)
f(x)
X max
X min
• f(x)
• f(x)
• X infl
y = ( x - 4)
2
3
7. 1
3
Sketch the graph of the equation y = 6x + 3x
•
•
•
•
•
•
•
•
•
x-intercept
y-intercept
VA
HA
SA
f(x)
f(x)
X max
X min
• f(x)
• f(x)
• X infl
4
3
