Wize High School Algebra II Textbook (Common Core) > Rational Functions

Graphs of Rational Functions

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Rational Functions
A rational function can be expressed as:
f(x)=axn+bxn−1+...cxm+dxm−1+...\boxed{f(x)=\displaystyle\frac{ax^n+bx^{n-1}+...}{cx^m+dx^{m-1}+...}}f(x)=cxm+dxm−1+...axn+bxn−1+...​​
where nnn is the largest exponent in the numerator and mmm is the largest exponent in the denomiator. 

Properties of Rational Functions
X-Intercepts can be found by setting f(x)=0f(x)=0f(x)=0 and solving for xxx
Vertical asymptotes can be found by setting the denominator to 0 and solving for xxx
Horizontal asymptotes:
If n<m‾\underline{n<m}n<m​: There is a horizontal asymptote at y=0y=0y=0
If n=m‾\underline{n=m}n=m​: There is a horizontal asymptote at y=acy=\displaystyle\frac{a}{c}y=ca​
If n>m‾\underline{n>m}n>m​: There is no horizontal asymptote. There may be an oblique/diagonal asymptote.
If factoring the numerator & denominator of f(x)f(x)f(x), any terms that cancel out indicates where a missing point/hole in f(x)f(x)f(x)
The domain is the set of all real numbers except where there are vertical asymptotes & missing points/holes
The range is the set of all real numbers except where there are horizontal asymptotes in most cases
Watch Out!
Rational functions may only attain values strictly above or below horizontal asymptotes. This will affect the range.

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Example 1

Let's look at f(x)=2x−3x+1f(x)=\displaystyle\frac{2x-3}{x+1}f(x)=x+12x−3​.




xxx-Intercept: (32,0)\Bigg(\displaystyle\frac{3}{2},0\Bigg)(23​,0)
Vertical Asymptote: x=−1x=-1x=−1
Horizontal Asymptote: y=2y=2y=2
Missing Points/Holes: None
Intervals of Increasing: (−∞,−1) ∪ (−1,∞)(-\infin,-1)~\cup~(-1,\infin)(−∞,−1) ∪ (−1,∞)
Intervals of Decreasing: NA
Domain: (−∞,−1) ∪ (−1,∞)(-\infin,-1)~\cup~(-1,\infin)(−∞,−1) ∪ (−1,∞)
Range: (−∞,2) ∪ (2,∞)(-\infin,2)~\cup~(2,\infin)(−∞,2) ∪ (2,∞)

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Example: Rational Functions
A graph of the function f(x)=x2−6x+8x2−3x−4f(x)=\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}f(x)=x2−3x−4x2−6x+8​ is sketched below:


Identify the equations for any vertical & horizontal asymptotes, and missing points/holes.

Factor f(x)f(x)f(x):

f(x)=x2−6x+8x2−3x−4f(x)=(x−4)(x−2)(x−4)(x+1)f(x)=x−2x+1\begin{array}{rcl}
f(x)&=&\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}\\\\
f(x)&=&\displaystyle{\frac{(x-4)(x-2)}{(x-4)(x+1)}}\\\\
f(x)&=&\displaystyle{\frac{x-2}{x+1}}

\end{array}f(x)f(x)f(x)​===​x2−3x−4x2−6x+8​(x−4)(x+1)(x−4)(x−2)​x+1x−2​​

Vertical Asymptotes: 

Let x+1=0x+1=0x+1=0. 

The vertical asymptote is at x=−1.x=-1.x=−1.

Horizontal Asymptote:

The degree of the numerator is equivalent to the degree in the denominator. 

Therefore, y=1y=1y=1.

Missing Points/Holes

Missing points/holes exist whenever terms in the numerator and denominator cancel. 

Since the term (x−4)(x-4)(x−4) was eliminated, there is a missing point/hole at x=4.x=4.x=4.

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Example: Rational Functions 
A graph of the function f(x)=x+5x2−4f(x)=\displaystyle{\frac{x+5}{x^2-4}}f(x)=x2−4x+5​ is sketched below:

Identify the equations for any vertical & horizontal or oblique asymptotes, and missing points/holes.

Factor f(x)f(x)f(x):

f(x)=x+5x2−4f(x)=(x+5)(x−2)(x+2)\begin{array}{rcl}
f(x)&=&\displaystyle{\frac{x+5}{x^2-4}}\\\\
f(x)&=&\displaystyle{\frac{(x+5)}{(x-2)(x+2)}}\\\\
\end{array}f(x)f(x)​==​x2−4x+5​(x−2)(x+2)(x+5)​​

Vertical Asymptotes: 

Let x−2=0x-2=0x−2=0 and let x+2=0.x+2=0.x+2=0.

The vertical asymptotes are at x=±2.x=\pm2.x=±2.

Horizontal Asymptote:

The degree of the numerator is less than the degree in the denominator. 

Therefore, y=0y=0y=0.

Missing Points/Holes

Missing points/holes exist whenever terms in the numerator and denominator cancel. 

Since there are no terms the cancel out, there are no missing points/holes.

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Example: Rational Functions 
A graph of the function f(x)=x2+x−12x+1f(x)=\displaystyle{\frac{x^2+x-12}{x+1}}f(x)=x+1x2+x−12​ is sketched below:

Identify the equations for any vertical & horizontal or oblique asymptotes, and missing points/holes.

Factor f(x)f(x)f(x):

f(x)=x2+x−12x+1f(x)=(x−3)(x+4)(x+1)\begin{array}{rcl}
f(x)&=&\displaystyle{\frac{x^2+x-12}{x+1}}\\\\
f(x)&=&\displaystyle{\frac{(x-3)(x+4)}{(x+1)}}
\end{array}f(x)f(x)​==​x+1x2+x−12​(x+1)(x−3)(x+4)​​

Vertical Asymptotes: 

Let x+1=0x+1=0x+1=0 .

The vertical asymptote is at x=1x=1x=1.

Horizontal Asymptote:

The degree of the numerator is greater than the degree in the denominator. 

Therefore, there is an oblique asymptote which can be found by dividing x2+x−12x+1\displaystyle\frac{x^2+x-12}{x+1}x+1x2+x−12​:

11−12−1↓−1010−12\begin{array}{r|ccc}
&1&1&-12\\
-1&\downarrow&-1&0\\\hline
&1&0&-12
\end{array}−1​1↓1​1−10​−120−12​​

The oblique asymptote does not count the remainder. 

So, the oblique asymptote becomes the quotient y=xy=xy=x.

Missing Points/Holes

Missing points/holes exist whenever terms in the numerator and denominator cancel. 

Since there are no terms the cancel out, there are no missing points/holes.

Practice: Rational Functions
Match the reciprocal function with the correct sketch of its graph. 

A.

B.

C.

f(x)=2x−74x+9f(x)=\displaystyle\frac{2x-7}{4x+9}f(x)=4x+92x−7​ 

f(x)=3x+42x−3f(x)=\displaystyle\frac{3x+4}{2x-3}f(x)=2x−33x+4​  

f(x)=3x2+x−42x2−5x+3f(x)=\displaystyle\frac{3x^2+x-4}{2x^2-5x+3}f(x)=2x2−5x+33x2+x−4​ 

I don't know

Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of: 

f(x)=x2−11x+18x2−3x+2f(x)=\displaystyle\frac{x^2-11x+18}{x^2-3x+2}f(x)=x2−3x+2x2−11x+18​

A) Vertical Asymptote

x=

B) Horizontal/Oblique Asymptote

y=

x

-Intercepts

x=

D) Missing Points (if there are none, put none)

x=

I don't know

Practice: Rational Functions
Match the equation for the rational function with its appropriate properties. 

A.

A vertical asymptote at x=−2x=-2x=−2  
A horizontal asymptote at y=0y=0y=0 

B.

A vertical asymptote at 92\frac{9}{2}29​    
An x-intercept at −53-\frac{5}{3}−35​  

C.

A missing point at x=−2x = -2x=−2 

D.

A vertical asymptote at 53\frac{5}{3}35​   
An x-intercept at 92\frac{9}{2}29​ 

f(x)=x2−4x+2f(x)=\displaystyle\frac{x^2-4}{x+2}f(x)=x+2x2−4​ 

f(x)=2x−9−3x+5f(x)=\displaystyle\frac{2x-9}{-3x+5}f(x)=−3x+52x−9​ 

f(x)=3x+52x−9f(x)=\displaystyle\frac{3x+5}{2x-9}f(x)=2x−93x+5​ 

f(x)=x−2x2−4f(x)=\displaystyle\frac{x-2}{x^2-4}f(x)=x2−4x−2​ 

I don't know

Extra Practice

Rational Functions

Let f(x)=x2+5x+6x−4f(x)=\dfrac{x^2+5x+6}{x-4}f(x)=x−4x2+5x+6​ be a rational function. 

Graphing Rational Functions

Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d}f(x)=cx+dax+b​ that describes the following:
Vertical asymptote at x=−1x=-1x=−1

Graphing Rational Functions

Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d}f(x)=cx+dax+b​ that describes the following:
Vertical asymptote at x=−2x=-2x=−2

Graphing Rational Functions

Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}f(x)=x+5x2−25​, finding all asymptotes, x-intercepts, missing points, and positive & negative intervals. 

Graphing Rational Functions

Practice: Graphing Rational Functions
Sketch a graph of f(x)=−x+9x+3f(x)=\dfrac{-x+9}{x+3}f(x)=x+3−x+9​, finding all asymptotes, intercepts, missing points, and positive & negative intervals. 

Rational Functions

Practice: Rational Functions
Match the rational function with the correct sketch of its graph. 

Rational Functions

Practice: Rational Functions
Match the rational function with the correct sketch of its graph. 