1AAsymptotes

= Asymptotes =
 * // Pronounced ass-im-tote (the p is silent)  //


 * An ** asymptote ** is a line which the graph approaches but never touches.


 * Drawing Graphs with Asymptotes **


 * Care should be taken when drawing a graph approaching an asymptote.
 * The line should approach but not cross or touch the asymptote.
 * The line should never curl away from the asymptote.




 * Finding the asymptotes for reciprocal graphs **


 * A __ **vertical asymptote** __ will be formed when the denominator of the fraction becomes zero
 * the original function has an x-intercept

math . \qquad \textbf{For example} \\.\\ . \qquad \text{The graph of } y=\dfrac{1}{x-6} math
 * will have a vertical asymptote where x – 6 = 0
 * so it will have a vertical asymptote where x = 6


 * Notice that x = 6 is the x-intercept of the line y = x – 6

Reciprocal graphs

math \\ . \qquad \text{All reciprocal graphs in the form } y = \dfrac{1}{f(x)} \\. \\ . \qquad \text{ have a } \textbf{horizontal asymptote} \text{ at } y=0 math


 * including both
 * hyperbola and
 * truncus


 * Oblique Aysmptotes **


 * More complex graphs may have an __ **oblique** __ (sloping) or __**curved**__ asymptote.


 * The equation of this asymptote can be determined by
 * covering the fraction part of the function with your hand.
 * The remainder is the equation of the other asymptote.

Example :


 * The following equation will produce a graph with two asymptotes.

math . \qquad \qquad y=\dfrac{2}{x-3}+2x-3 math


 * To find the vertical asymptote, only consider the fraction part.

math . \qquad \qquad \dfrac{2}{x-3} math


 * This will be undefined when the denominator equals zero
 * which occurs when // x // = 3


 * So the vertical asymptote is at // x // = 3


 * To find the other asymptote,
 * cover the fraction and only consider the remaining part.


 * This gives // y // = 2 // x // – 3.


 * So // y // = 2 // x // – 3 is the equation of the other asymptote.
 * In this case it is an oblique (sloping) asymptote.

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