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A Summary of Curve
Sketching
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)
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)
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
Let’ make a list
•
•
•
•
•
•
•
•
•

-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
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
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
5.2 first and second derivative test

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5.2 first and second derivative test

  • 1. A Summary of Curve Sketching
  • 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 • • • • • • • • • -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