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 : translates function left or right.
 Moves the vertical asymptote.
 Vertical asymptote is the line x = h; D: {x | x ≠ h}
◦ k : translates function up or down.
 Moves the horizontal asymptote.
 Horizontal asymptote is the line y = k: R: {y |y ≠ k}
Vertical Asymptote
Set the Denominator of
the function to zero and
solve.
 x – 4 = 0; x-4+4=0+4;
x=4
 The Vertical Asymptote
is the vertical line with
equation x=4 and the
Domain is {x x≠4}

3
f (x) =
+5
x-4
Note: When graphing
choose 3 x-values to
the LEFT of the
Vertical Asymptote and
3 x-values to the
RIGHT.
Horizontal Asymptote

To find the horizontal asymptote if the
numerator is a constant then the Horizontal
Asymptote is y=0, otherwise set the term
with the greatest degree of the numerator
over the term with the greatest degree in
the denominator and reduce (simplify the
terms).
2x
f (x) =
-6
3
x-7
f (x) =
+5
x-4
The numerator is a constant,
3, therefore the HA is y=0 and
the Range is {y y≠0}.
The numerator term with greatest
degree is 2x and the denominator
term with the greatest degree is x
set them in a new fraction and
simplify, y = 2x = 2 = 2 the Range is
x 1
{y y≠2}.
Holes in the Graph

When a factor in
numerator cancels
with a factor in the
denominator
4x 2 + x x(4x +1) (4x +1)
=
=
2
2x + x x(2x +1) (2x +1)

Set the factor that
cancelled = 0 and
solve. x=0 is the
hole on the graph.
Slant Asymptotes
Watch this YouTube video to find out how.
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 and 6 units down
from the parent function.
V. asymptote: x = 3
H. asymptote: y = -6
Plot anchor points and sketch
the function.
Asymptote Review
Vertical Asymptotes-set denominator = 0.
Horizontal Asymptotes:
1. Constant in numerator-HA is y=0.
2. Degree is equal in numerator and
denominator y = reduced answer.
3. The degree of the numerator is one smaller
than the degree of the denominator, then
y = 0 (since the top will simplify to a
constant).
4. Degree in the numerator is one bigger than
the degree in the denominator then
y = (slant asymptote equation).

Homework
Textbook Section 3-7(pg. 186): 14-23, 24-28 even
Extra Credit: page 187 30-40 done in their entirety,
all work shown and graphs drawn (x-values 3 to the
left and 3 to the right of VA).
Exit Ticket
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.