Section 4.3 – Curve Fitting
We will be learning how to sketch the graph of a function by using application of the 1st and 2nd derivatives.
Vertical Asymptotes:
A function, f , has a vertical asymptote at the line x  a , if:

lim f  x    or 
x a 
or
lim f  x    or 
x a 
What we will do to find vertical asymptotes is:
1. Factor and reduce the function if possible
2. Set the denominator equal to zero and solve for x. Any REAL NUMBERS you obtained is where the function has
a vertical asymptote.
f  x 
1
x
Horizontal Asymptotes:
A function, f , has a horizontal asymptote at the line y  b , if:

lim f  x   b
x 
or
lim f  x   b
x 
To find the horizontal asymptote:
1. Take the limit as the function approaches infinity.
f  x 
1
x
Find the vertical and horizontal asymptotes of the following functions:
EX1:
EX2: y 
1
x3
t2
EX4: g  t   2
t 9
EX3: y 
x
x 1
2
EX5: f  x   2 
5
 x  2
2
To sketch the curve of a function:
1. Find the domain of the function
2. Find the x and y intercepts of the function
3. Polynomial Functions: Determine the end behavior of x (behavior of f for large absolute values of x)
4. Rational Functions: Find all the horizontal and vertical asymptotes of the function
Note: The graph of a polynomial function does NOT have asymptotes
5. Determine intervals of increasing and decreasing
6. Find the relative extrema
7. Determine concavity
8. Find the inflection points
9. Plot all the information above and any additional points necessary and sketch the graph.
Sketch the graph of the following functions:
EX6:
EX7:
EX8: 𝑓(𝑥) = 𝑥 4 − 2𝑥 2 − 3
Domain: (−∞, ∞)
X-intercept(s): (√3, 0) and (−√3, 0)
y-intercept: (0, −3)
Intervals of Inc/Dec
Increasing: (−1, 0) ∪ (1, ∞)
Decreasing: (−∞, −1) ∪ (0,1)
Rel. min. (−1, −4) (1, − 4)
Rel. max. (0, −3)
1
1
Concave up: (−∞, −√3) and (√3 , ∞)
1
1
Concave down: (−√3 , √3)
1
3
Inflection points: (−√ , −
32
) and
9
1
(√3 , −
EX9:
Domain
(−∞, −1) ∪ (−1, ∞)
Intercepts (1, 0) (0, −1)
Vertical Asymptote: 𝑥 = 1
Horizontal Asymptote: y  1
Intervals of Inc/Dec
Inc: (−∞, −1) ∪ (−1, ∞)
Dec: NONE
Rel Max.: NONE; Rel Min: NONE
Concaved Down on: (−1, ∞)
Concaved Up on : (−∞, −1)
Point of Inflection NONE
32
)
9
EX10: f  t  
t2
t2  9
Domain:__________________________
a) X-intercept(s):_____________________
y-intercept:_______________________
b) Horizontal Asymptote:________________
Vertical Asymptote:__________________
c) Increasing:________________________
decreasing:________________________
d) Relative Extrema:___________________
e) Concave up:________________________
Concave down:________________________
f)
Inflection points:_______________________