Unit 7 – Rational Functions
Topic: Transformations of the Rational Parent
Function
Rational Parent Function
1
f ( x) 
x
Graph of the rational
parent function is a
hyperbola.
 Vertical asymptote at
x = 0; D: {x | x ≠ 0}
 Horizontal asymptote
at y = 0; R: {y | y ≠ 0}
 Asymptote: boundary
line for the graph of
the function.

Transforming the Rational Parent Function
a
f ( x) 
k
xh
General format of a rational function:
 Possible transformations (we’ve done this before):

◦
◦
◦
◦
|a| > 1: stretches hyperbola away from origin.
|a| < 1: compresses hyperbola towards origin.
a < 1: reflects graph across x-axis.
h (“bad child”): translates function left or right.
 Moves the vertical asymptote.
 Vertical asymptote is the line x = h; D: {x | x ≠ h}
◦ k (“good child”): translates function up or down.
 Moves the horizontal asymptote.
 Horizontal asymptote is the line y = k: R: {y |y ≠ k}
Transforming the Rational Parent Function

Identify the asymptotes, domain & range for the given
function, then sketch the graph of the function.
1
f ( x) 
4
x2



V. asymptote: x = –2 (remember to
change the sign for h)
H. asymptote: y = 4
D: {x | x ≠ –2}; R: {y | y ≠ 4}
• Plot asymptotes
• Since everything shifted left 2
& up 4, the points (1, 1) & (–
1, –1) from the parent
function are now (–1, 5) & (–
3, 3). Plot these points.
• Sketch the resulting hyperbola
through those points.
Transforming the Rational Parent Function

Using the rational parent function as a guide, describe
the transformations and graph the function.
1
f ( x) 
6
x 3




The function will translate 3
units right (“bad child”) and 6
units down (“good child”)
from the parent function.
V. asymptote: x = 3
H. asymptote: y = -6
Plot anchor points and sketch
the function.
Journal Entry
TITLE: Rational Functions 3-2-1
Identify 3 things you already knew from the Powerpoint,
2 new things you learned, and one question you still have.
Homework
Textbook Section 8-4 (pg. 597): 2-7, 17-22
Due 2/24