Alg II 8­3 Rational Functions and Their Graphs ­ End.notebook
November 15, 2016
8-3 Rational Functions and Their Graphs
A rational function is a function that you can write in the form
where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those values for which Q(x) = 0.
Example:
Continuity
A graph is continuous when it has no jumps, breaks, or holes. You can draw the graph and your pencil never leaves the paper.
Graphs that do have a jump, break, or hole are called discontinuous graphs. A point where the graph is not continuous is called a point of discontinuity.
There are two types of discontinuity:
1.
Removable Discontinuity ⇒ If you reduce the function, the discontinuity will disappear. This is type of discontinuity is also called a "hole" in the graph. 2.
Non­removable discontinuity ⇒ You cannot reduce the function in a way that makes the discontinuity disappear. This is type of discontinuity is also called a vertical asymptote.
1
Alg II 8­3 Rational Functions and Their Graphs ­ End.notebook
November 15, 2016
Ex. 1 What are the points of discontinuity? Are they removable or non­removable?
A.
B.
C.
Vertical Asymptotes and Holes
A vertical asymptote is an invisible vertical line which the graph will approach, but will never touch or cross. x = −1
To find the vertical asymptote of a rational function, you set the denominator equal to zero, then solve for x.
A hole is a single point at which the function has no value. A hole is formed when a value of x sets both the denominator and the numerator of a rational function equal to 0.
2
Alg II 8­3 Rational Functions and Their Graphs ­ End.notebook
November 15, 2016
Ex. 2 Find the vertical asymptotes and holes for each rational function.
A.
B.
C.
Horizontal Asymptotes
Just like a vertical asymptote, a horizontal asymptote is an invisible horizontal line which the graph will approach, but will never touch or cross.
y = 1
A horizontal asymptote is the line formed from the coefficients of the highest exponent of the function.
3
Alg II 8­3 Rational Functions and Their Graphs ­ End.notebook
November 15, 2016
Ex. 3 Find the horizontal asymptote of each rational function.
A.
B.
C.
Ex. 4 Sketch the graph of each rational function.
A.
y
horizontal asymptote
vertical asymptote
hole
x
x­intercepts
y­intercepts
extra points
B.
y
horizontal asymptote
vertical asymptote
hole
x
x­intercepts
y­intercepts
extra points
4