5.4A Rational Functions
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities; and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context.
For the Board: You will be able to graph rational functions and transform rational functions by
changing the parameters.
Bell Work 5.4:
Find the zeros of each function.
1. f(x) = x2 + 2x – 15
2. f(x) = x2 – 49
Simplify. Identify any x-values for which the expression is undefined.
Hint: a rational expression is undefined when the denominator equals zero.
x  5x  4
x2  1
2
3.
4.
x 2  8 x  12
x 2  12 x  36
Anticipatory Set:
A rational function is a function whose rule can be written as a ratio of two polynomials.
1
The rational parent function is f(x) = .
x
x
y
-2
-½
-1
-1
-½
-2
½ 1
2 1
2
½
It’s graph is a hyperbola with two separate branches.
It has two asymptotes. Lines that the graph approaches
but never reaches.
Vertical asymptote: x = 0
Horizontal asymptote: y = 0.
Most rational functions are discontinuous functions having one or more gaps or breaks.
All quadratic and polynomial functions are continuous function having no gaps or breaks.
1
can be translated in the same way that quadratics are translated.
x
1
Vertical translations: f(x) = x2 + 2
f(x) = + 2
x
1
f(x) = x2 – 2
f(x) = – 2
x
Horizontal translations:
1
f(x) = (x – 2)2
f(x) =
x2
The function f(x) =
f(x) = (x + 2)2
f(x) =
1
x2
f(x) =
3
x
f(x) =
1
3x 
Vertical stretch/compression:
f(x) = 3x2
Horizontal stretch/compression:
f(x) = (3x)2
If the function is translated up/down, then the horizontal asymptote is translated up/down and the
vertical asymptote remains unchanged.
If the function is translated right/left, then the vertical asymptote is translated right/left and the
horizontal asymptote remains unchanged.
f(x) =
a
k
xh
a: vertical stretch or compression factor.
0<a<1
compression
a>1
stretch
a negative
“flip” over the x-axis
h: horizontal translation (reverse sign)
vertical asymptote: x = h
domain: {x|x ≠ h}
k: vertical translation
horizontal asymptote: y = k
range: {y|y ≠ k}
Open the book to page 340 and read example 1.
1
Example: Using the graph of f(x) = as a guide, describe the transformation and graph each function.
x
State both the horizontal and vertical asymptote.
1
Graph f(x) = then shift the key ordered pairs.
x
1
1
a. g(x) =
b. g(x) =  3
x2
x
translation 2 units left
translation 3 units down
horizontal asymptote: y = 0
horizontal asymptote: y = -3
vertical asymptote: x = -2
vertical asymptote: x = 0
Graphing Activity:
1
as a guide, describe the transformation and graph each function.
x
State both the horizontal and vertical asymptote.
1
1
a. g(x) =
b. g(x) =  1
x4
x
translation left 4 units
translation up 1 unit
horizontal asymptote: y = 0
horizontal asymptote: y = 1
vertical asymptote: x = -4
vertical asymptote: x = 0
Practice: Using the graph of f(x) =
Open the book to page 341 and read example 2.
Example: Identify the asymptotes, domain, and range of the function g(x) =
Asymptotes: x = -3 and y = -2
Domain: {x|x ≠ -3}
1
2.
x3
Range: {y|y ≠ -2}
White Board Activity:
Practice: Identify the asymptotes, domain, and range of the function g(x) =
Asymptotes: x = 3 and y = -5
Domain: {x|x ≠3}
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 345 prob. 2 – 7, 17 – 22.
For a Grade:
Text: pg. 345 prob. 18, 20.
Range: {y|y ≠ -5}
1
 5.
x3