Aim: What concepts have we available to
aide us in sketching functions?
Do Now:
Find the domain of f  x  
Aim: Curve Sketching
2  x 2  9
x2  4
Course: Calculus
Concepts used in Sketching
•
•
•
•
•
•
•
•
•
•
x- and y-intercepts
symmetry
domain & range
continuity
vertical asymptotes
differentiability
relative extrema
concavity
points of inflection
horizontal asymptotes
Use them all? If not all, which are best?
Aim: Curve Sketching
Course: Calculus
Guidelines for Analyzing Graph
1. Determine the domain and range of the
function.
2. Determine the intercepts and asymptotes
of the graph.
3. Locate the x-values for which f’(x) and
f’’(x) are either zero or undefined. Use the
results to determine relative extrema and
points of inflection.
Also helpful: symmetry; end behavior
Aim: Curve Sketching
Course: Calculus
Abridged Guidelines – the 4 Tees
T1 Test the function
T2 Test the 1st Derivative
T3 Test the 2nd Derivative
T4 Test End Behavior
Aim: Curve Sketching
Course: Calculus
Model Problem 1
Analyze the graph of f  x  
2  x 2  9
x 4
2
1. find domain & range
exclusions at zeros of denominator
4
domain: all reals except ±2
2
-5
5
-2
-4
Aim: Curve Sketching
Course: Calculus
Model Problem 1
Analyze the graph of f  x  
2  x 2  9
x 4
2
2. find intercepts & asymptotes
y-intercept
f  0 
2  x 2  9
4
2

x 4
2  02  9 
2
02  4
-5
x-intercept
18 9


4 2
2  x 2  9
5
f  x  0 
-4
0
Aim: Curve Sketching
x2  4
-2
2  x 2  9
x 4
2
,
x  3
Course: Calculus
Model Problem 1
Analyze the graph of f  x  
2  x 2  9
x 4
2
2. find intercepts & asymptotes
verticals asymptotes found
at zeros of denominator
x = ±2
4
2
horizontal asymptote
If degree of p = degree of q, then the line
y = an/bm is a horizontal asymptote.
-5
5
-2
2  x 2  9
x 4
2
2 x  18
 2
x 4
2
-4
Aim: Curve Sketching
y=2
Course: Calculus
Model Problem 1
Analyze the graph of f  x  
2  x 2  9
x 4
3. find f’(x) = 0 and f’’(x) = 0 or undefined
f ' x  
2
2
2
x

4
4
x

2
x

   18  2 x 
x
4
2
 4
2
2
f ' x  
x
-5
20 x
2
 4
2
0
x=0
5
-2


(x2 2– 4)2 = 0 2
2 x 9
2 x  18
 at 2zeros
undefined
2
x 4
x 4
of denominator
Aim: Curve Sketching
-4
x = ±2
Course: Calculus
Model Problem 1
2 20 x
2
x
 9

f
'
x



2
Analyze the graph of f  x  
2 x2  4
x  4 
3. find f’(x) = 0 and f’’(x) = 0 or undefined
f ''  x  
f ''  x  
2
1
x

4
20

20
x
2(
x

4)


 2x 



2
2

20  3 x 2  4 
x
2
 4
3
2
x
  4
0

2 2
no real
solution
no possible points of inflection
Aim: Curve Sketching
Course: Calculus
Model Problem 1
3. test intervals
f(x)
f’(x)
f’’(x)
characteristic of
Graph
decreasing,
concave down
- < x < -2
Undef Undef Undef vertical asymptote
decreasing,
-2 < x < 0
+
concave up
x = -2
x=0
9/2
0
+
relative minimum
increasing,
0<x<2
+
+
concave up
x=2
Undef Undef Undef vertical asymptote
increasing,
Aim: Curve Sketching
Course: Calculus
+
2<x<
concave down
Model Problem 1
q x  =
2x 2-9
x 2-4
6
decreasing,
increasing,
concave up
concave up
(0, 9/2)
-2 < x < 0 relative minimum 0 < x < 2
4
2
-5
increasing,
concave down
- < x < -2
5
-2
-4
increasing,
concave down
2<x<
-6
Aim: Curve Sketching
Course: Calculus
Model Problem 2 – What the cusp!!
Analyze the graph of y  2  x
2
3
T1
Find Domain
all reals
Find intercepts & asymptotes
0  2 x
2
3
 x
2
3
 
 2
x
2
3
3
2
  2
x -intercepts at  2 2
y -intercepts at (0, 2)
no vertical or
horizontal asymptotes
Aim: Curve Sketching
Course: Calculus
3
2
Model Problem 2 – What the cusp!!
Analyze the graph of y  2  x
2
3
T2
1st Derivative Test

1
3
dy
2
2
 x
 x
dx
3
3
BUT . . .
x = 0 is defined for
original function

1
3
x at 0 is
0
undefined
f’ > 0
inc
h x  =
3
2
2-x 3
f’ < 0
dec
3
2
2
a cusp!!!
1
1
-2
g x  = -

2
3
-1
x 3
2
-1
Aim: Curve Sketching
-2
-2
2
-1
Course: Calculus
-2
Model Problem 2 – What the cusp!!
Analyze the graph of y  2  x
2
3
T3
2nd Derivative Test
x at 0 is
d 2 y 2  43
2  43
 x
x 0
2
undefined
dx
9
9
f’’ > 0
f’’ > 0
con up
con up
h x  =
2
2-x 3
3
cusp
2
1
-2
2
-1
Aim: Curve Sketching
-2
Course: Calculus
Model Problem 3
3 x 5  20 x 3
Sketch the graph f ( x ) 
32
<0
<0
>0
>0
f’
dec
dec
inc
inc
>0 <0
>0
<0
f’’
c.u. c.d.
c.u.
c.d.
f x  =
3x 5-20x 3
3
32
(-2,2)
relative
max.
-4
2
1
-2
2
4
-1
inflection points:
(-1.4,1.2), (0,0), (1.4,-1.2)
-2
Aim: Curve Sketching
-3
relative
(2,-2)
Course: Calculus
min.
Model Problem 4
Analyze the graph of
cos x
f  x 
1  sin x
1. find Domain
2. find intercepts & asymptotes
verticals asymptotes found
a zeros of denominator
x = ±2
1 + sin x = 0; sin x = -1 x  
Aim: Curve Sketching

2
Course: Calculus