Post on 28-Feb-2020
5.3 Graphing General Rational Functions.notebook May 11, 2015
Bell Work
Factor the following expressions.
1) 12x2 + 5x 2 2) x2 5x 50
5.3 Graph General Rational FunctionsIn section 5.2 we learned how to graph simple rational functions.
Two Forms:
1. y = ax h + k VA:
HA:
2. y = ax + bcx + dVA:
HA:
5.3 Graphing General Rational Functions.notebook May 11, 2015
To graph general rational functions we will find 5 things:
* xintercept(s)
* vertical asymptote(s)
* horizontal asymptote
* hole
* point of discontinuity
Points of discontinuity
are moments within a function that are undefined and appear as a break or hole in a graph
created when a function is presented as a fraction and an inputted variable creates a denominator equal to zero
Discontinuities: 1, 3
x intercepts: (0,0), (4,0)
Vertical Asym: x = 1, x=3
Horizontal Asym: y=3
Holes: None
Example
5.3 Graphing General Rational Functions.notebook May 11, 2015
Discontinuities: 1, 3
x intercepts: (0,0), (4,0)
Vertical Asym: x = 1
Horizontal Asym: none
Holes: (3, 1.5)
Example
x intercept(s)Find the xvalues that make the numerator zero.
Example 1: Find the xintercepts of the function.
x2 + 3x 4y = x 2
_________
5.3 Graphing General Rational Functions.notebook May 11, 2015
Vertical Asymptotes*Find the xvalues that make the denominator zero.
Example 2: Find all the vertical asymptotes ofa. f(x) =
xx2 + 5x + 6
b. f(x) = x2 4x 12
x2 11x + 30_____________
Horizontal Asymptotes*Find the degree of the numerator and denominator.
Case 1 m < nThe line y = 0
is a horizontal asymptote.
Case 2 m = nThe line y =
a/c is a horizontal asymptote.
Case 3 m > nThe graph has no
horizontal asymptote.
y = axm
cxn ___
5.3 Graphing General Rational Functions.notebook May 11, 2015
Example 3: Find all the vertical and horizontal asymptotes.
2x2 2x + 1 x2 x 12f(x) =
vertical:
horizontal:
HolesIf x b is a factor of both the numerator and
denominator then there is a hole in the graph when x = b unless it is a vertical asymptote.
3x2 + x3 x2 + 2x 3
f(x) =
HA:
VA:
Hole:
Example 4: Identify all the asymptotes and holes
in the graph.
5.3 Graphing General Rational Functions.notebook May 11, 2015
Graph the function.1. Find all the xintercepts, asymptotes and holes, and graph them. 2. Use the graphing calculator tofind some more points.
2x 1
x + 4y =Example 5
xintercept(s):
Vertical Asymptote:
Horizontal Asymptote:
Holes:
D:
R:
y = 3x 28 + x2
x2 + 12x + 35
Example 6
xint:
VA:
HA:
Holes:
D:
R:
5.3 Graphing General Rational Functions.notebook May 11, 2015
Example 7
xint:
VA:
HA:
Holes:
D:
R:
y = x2 +1 x2 1
Example 8
xint:
VA:
HA:
Holes:
D:
R:
y = 4x2 5x + 4 _________
5.3 Graphing General Rational Functions.notebook May 11, 2015
Assignment: p. 322 # 8, 10, 12, 16, 22