Ch 2.6 & Ch 2.7 - Rational Functions and Aymptotes & The Graphs of Rational Functions

Chapter 2.6 - Rational Functions and Asymptotes

 

A rational function can be written in the form:

f(x) = N(x)/D(x)

 

where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

 

 

To find the x and y intercepts:

When f(x), or even N(x), equals 0, the equation will yield its x-intercepts.

When x equals 0, otherwise written as f(0), the equation will yield its y-intercept.

 

To find the horizontal and vertical asymptotes:

When D(x) equals 0, it will yield an "undefined solution," providing the vertical asymptote.

To find the horizontal asymptote, one must first identify the equation's End Behavior.

 

For example, in the equation:

f(x) = ( 3x (x + 4)) / ( (x-5)(x+1) )



3 is the leading term of N(x); therefore, the horizontal asymptote falls at x = 0.

 

 

Chapter 2.7 - Graphs of Rational Functions

 

When graphing a rational function, locating the intercepts and asymptotes are crucial.

 

For example, the equation:

f(x) = (x^2 + x - 6) / (x^2 - x - 6)

 must be broken down to locate these graphing points.

 

To find the x-intercepts, the numerator, (x^2 + x - 6), must be set equal to 0.

-After factoring, the solutions that are left are x = 2 and x = -3

To find the y-intercept, the value of x must be set to 0.

-After solving the equation, the y-intercept is y = 1

 

To locate the vertical asymptote, the denominator, D(x), must be set to equal 0.

-The factored denominator, (x^2 - x - 6), yields the values of x: x = 3 and x = -2, the vertical asymptotes

The horizontal asymptote, as previously stated, identifies with the equation's leading coefficient, or, End Behavior

-In this equation, the leading coefficient is 1, so, the horizontal asymptote = 1.

 

All in all,

x-intercepts: (2,0) and (3,0)

y-intercept: (0,1)

Vertical Asymptotes: x = 3 and x = -2

Horizontal Asymptote: y = 1