Rational Functions - Revision history

MathAdmin: /* Asymptotes */

2015-10-20T04:32:04Z

Asymptotes

MathAdmin at 04:31, 20 October 2015

2015-10-20T04:31:15Z

MathAdmin: Created page with "
TOC
==Introduction== Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominat..."

2015-10-20T04:22:59Z

Created page with "<div class="noautonum">__TOC__</div> ==Introduction== Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominat..."

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<div class="noautonum">__TOC__</div>
==Introduction==

Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominator is zero. In this section we will
focus on the algebraic aspects of rational functions. These aspects include vertical, horizontal, and oblique asymptotes.

==Definition==

A rational function is a function of the form <math>R(x) = \frac{P(x)}{Q(x)}</math>, where Q(x) is not the zero polynomial.
The domain of a rational function is all real numbers except for the zeros of Q(x).

Example:

<math>\frac{x^2 + 3}{3x^4 - 5x^3 + 9}</math>

==Asymptotes==

An asymptote is a notion that is more widely applicable than to just rational functions.

Let R(x) be a function.
If as <math>x \rightarrrow \infty, \text{ or }x \rightarrrow -\infty</math> R(x) approaches some value L, then the line y = L is a horizontal asymptote of the graph
of R. For now just accept this as the definition. We will make this idea of R(x) approaching some value L a bit more concrete later on.

If, as x approaches some number c, the values <math>\vert R(x)\vert \rightarrrow \infty</math>, then the line x = c is a vertical asymptote of the graph of R.
This will get mentioned again later, but for rational functions vertical asymptotes correspond to zeros of the denominator that have higher multiplicity than
they do in the numerator, if they are even zeros of the numerator.

Theorem: Locating Vertical Asymptotes

Let <math>R(x) = \frac{P(x)}{Q(x)}</math> be a rational function in lowest terms, there are no common factors of P(x) and Q(x). Then the vertical asymptotes
of R(x) will occur at the zeros of Q(x). So, if r is a zero fo Q(x), then x = r will be a vertical asymptote.

Finding a horizontal aysmptote:

Let <math>R(x) = \frac{P(x)}{Q(x)} = \frac{a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0}{b_mx^m + b_{m-1}x^{m-1} + \ldots + b_1x + b_0}</math>

1) If <math> n < m</math>, then y = 0 is a horizontal asymptote.

2) If <math> n = m, \text{ then } y = \frac{a_n}{b_m}</math> is a horizontal asymptote

3) No horizontal asymptote.

[[Math_5|'''Return to Topics Page]]

@@ Line 26: / Line 26: @@
 they do in the numerator, if they are even zeros of the numerator.
-Theorem: Locating Vertical Asymptotes
+'''Theorem:''' Locating Vertical Asymptotes
 Let <math>R(x) = \frac{P(x)}{Q(x)}</math> be a rational function in lowest terms, there are no common factors of P(x) and Q(x). Then the vertical asymptotes

@@ Line 19: / Line 19: @@
 Let R(x) be a function.
-If as <math>x \rightarrrow \infty, \text{ or }x \rightarrrow -\infty</math> R(x) approaches some value L, then the line y = L is a horizontal asymptote of the graph
+If as <math>x\rightarrow\infty, \text{ or }x \rightarrow-\infty</math> R(x) approaches some value L, then the line y = L is a horizontal asymptote of the graph
 of R. For now just accept this as the definition. We will make this idea of R(x) approaching some value L a bit more concrete later on.
-If, as x approaches some number c, the values <math>\vert R(x)\vert \rightarrrow \infty</math>, then the line x = c is a vertical asymptote of the graph of R.
+If, as x approaches some number c, the values <math>\vert R(x)\vert \rightarrow\infty</math>, then the line x = c is a vertical asymptote of the graph of R.
 This will get mentioned again later, but for rational functions vertical asymptotes correspond to zeros of the denominator that have higher multiplicity than
 they do in the numerator, if they are even zeros of the numerator.

Rational Functions - Revision history

MathAdmin: /* Asymptotes */

MathAdmin at 04:31, 20 October 2015

MathAdmin: Created page with "__TOC__ ==Introduction== Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominat..."

MathAdmin: Created page with "
TOC
==Introduction== Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominat..."