Analyzing Rational Functions

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x f(x) = (3x² + 4x + 2)/(2x + 1)
-10 (3(-10)² + 4(-10) + 2)/(2(-10) + 1) = -13.7894736842105
-9 (3(-9)² + 4(-9) + 2)/(2(-9) + 1) = -12.2941176470588
-8 (3(-8)² + 4(-8) + 2)/(2(-8) + 1) = -10.8
-7 (3(-7)² + 4(-7) + 2)/(2(-7) + 1) = -9.30769230769231
-6 (3(-6)² + 4(-6) + 2)/(2(-6) + 1) = -7.81818181818182
-5 (3(-5)² + 4(-5) + 2)/(2(-5) + 1) = -6.33333333333333
-4 (3(-4)² + 4(-4) + 2)/(2(-4) + 1) = -4.85714285714286
-3 (3(-3)² + 4(-3) + 2)/(2(-3) + 1) = -3.4
-2 (3(-2)² + 4(-2) + 2)/(2(-2) + 1) = -2
-1 (3(-1)² + 4(-1) + 2)/(2(-1) + 1) = -1
0 (3(0)² + 4(0) + 2)/(2(0) + 1) = 2
1 (3(1)² + 4(1) + 2)/(2(1) + 1) = 3
2 (3(2)² + 4(2) + 2)/(2(2) + 1) = 4.4
3 (3(3)² + 4(3) + 2)/(2(3) + 1) = 5.85714285714286
4 (3(4)² + 4(4) + 2)/(2(4) + 1) = 7.33333333333333
5 (3(5)² + 4(5) + 2)/(2(5) + 1) = 8.81818181818182
6 (3(6)² + 4(6) + 2)/(2(6) + 1) = 10.3076923076923
7 (3(7)² + 4(7) + 2)/(2(7) + 1) = 11.8
8 (3(8)² + 4(8) + 2)/(2(8) + 1) = 13.2941176470588
9 (3(9)² + 4(9) + 2)/(2(9) + 1) = 14.7894736842105
10 (3(10)² + 4(10) + 2)/(2(10) + 1) = 16.2857142857143

                    

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Table Values:      Start:      End:      Stepsize:   

Draw horizontal asymptote y =

Draw vertical asymptote x =

Draw slant asymptote y =



Degree of polynomial in the numerator is 2

Degree of polynomial in the denominator is 1

Horizontal Asymptote:

Since the degree of the polynomial in the numerator is greater than the degree
of polynomial in the denominator, there will no horizontal asymptote.
In other words, as x approaches infininity (x → ∞), y = f(x) also approaches infinity (∞).


Vertical Asymptote:

The following statements about vertical asymptote are based on the assumption that 3x² + 4x + 2 and 2x + 1
do not have common factor(s). If 3x² + 4x + 2 and 2x + 1 have common factors, then
not all real solutions of the equation 2x + 1 = 0 will lead to vertical asymptotes.

To find vertical asymptote(s) set 2x + 1 = 0 and then solve for x.
If the equation 2x + 1 = 0 has real solution(s) then there will be vertical asymptote(s).
If the equation 2x + 1 = 0 has only nonreal solutions, then there will be no vertical asymptote.

2x + 1 = 0 has the following solution(s):

solution: x = -1/2



Vertical Asymptote is the vertical line x = -1/2


Slant Asymptote:

Since the degree of the polynomial in the numerator is one degree higher than the degree
of polynomial in the denominator, there will be a slant asymptote.

Note: (3x² + 4x + 2)/(2x + 1) = (3/2)x + (5/4) + [(3/4)]/[2x + 1]

Slant Asymptote is y = (3/2)x + (5/4)


Domain of f(x) = (3x² + 4x + 2)/(2x + 1):

To find the domain of the rational function, we set 2x + 1 = 0 and then solve for x.

If the equation 2x + 1 = 0 has real solution(s) then we need to exclude these real
values from the domain.
If the equation 2x + 1 = 0 has only nonreal solutions or the denominator is nonzero constant,
then the domain will be the set of all real numbers (-∞, ∞).

2x + 1 = 0 has the following solution(s):

x = -0.5



Domain = set of all real numbers except -0.5.

Domain = {x | x ≠ -0.5}





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Absolute Value and Vertical Bars:

For each absolute value expression, make sure that a matching pair of vertical bars are used.
When vertical bars are used to denote absolute value, this calculator is not designed to handle
the case of absolute value expression nested inside another absolute value expression because it's too
ambiguous to interpret user's intention. For a function like f(x) = | 2 + |x + 3| - x |, use "Abs".
f(x) = | 2 + |x + 3| - x | can be input as f(x) = Abs(2 + Abs(x+3) - x).


Links to other pages:

Instruction Manual on how to use graphing calculator

Scientific Calculator

Graphing Calculator with One Function

Graphing Calculator with Two Functions

Graphing Calculator with Three Functions

Graphing Calculator with Four Functions

Graph x = g(y)

Graphing Exponential Function: y = bx and y = ex