document.write( "Question 983779: Please help me with the equation below:\r
\n" ); document.write( "\n" ); document.write( "Identify the intercepts, asymptotes, and \"holes\" on this rational equation:\r
\n" ); document.write( "\n" ); document.write( "f(x) = x^3+4x^2-5x/x^2-1 \n" ); document.write( "
Algebra.Com's Answer #604575 by KMST(5328)\"\" \"About 
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I assume you mean \"f%28x%29=%28x%5E3%2B4x%5E2-5x%29%2F%28x%5E2-1%29\" .
\n" ); document.write( "The y-intercept is the point where the graph crosses the y-axis,
\n" ); document.write( "and for all points on the y-axis \"x=0\" ,
\n" ); document.write( "so the y-intercept is \"f%280%29=0%2F%28-1%29=0\" .
\n" ); document.write( "The function crosses the y-axis at the point with \"system%28x=0%2Cy=0%29\" ,
\n" ); document.write( "The \"highlight%28y-intercept%29\" is the point (0,0), the origin.
\n" ); document.write( "Factoring numerator and denominator, we get
\n" ); document.write( " .
\n" ); document.write( "For \"x%3C%3E1\" , the function is not defined, because the denominator is zero.
\n" ); document.write( "However for values of \"x\" around \"x=1\" \"f%28x%29\" has a value.
\n" ); document.write( "There is just a \"highlight%28hole%29\" at \"x=1\" .
\n" ); document.write( "Otherwise, for \"x%3C%3E1\" , the expression simplifies to
\n" ); document.write( "\"f%28x%29=x%28x%2B5%29%2F%28x%2B1%29\" ,
\n" ); document.write( "When a factor in the \"x%28x%2B5%29\" numerator is zero, \"f%28x%29=0\" ,
\n" ); document.write( "and the graph crosses the x-axis, with \"y=0\"
\n" ); document.write( "The points where that happens are the \"highlight%28x-intercepts%29\" :
\n" ); document.write( "\"highlight%28x=0%29\" , and
\n" ); document.write( "\"x%2B5=0\"--->\"highlight%28x=-5%29\" .
\n" ); document.write( "So the graph crosses the x-axis at (0,0) (we knew that point) and at (-5,0).
\n" ); document.write( "Also, for \"x%3C%3E1\" , we see that \"f%28x%29=x%28x%2B5%29%2F%28x%2B1%29\" is undefined for \"x=-1\" , because the denominator is zero.
\n" ); document.write( "For values of \"x\" very close to \"x=-1\" ,
\n" ); document.write( "the numerator, the quadratic polynomial , and
\n" ); document.write( "as \"x\" approaches \"-1\" , \"P%28x%29\" approaches \"-4\" .
\n" ); document.write( "On the other hand, as \"x\" approaches \"-1\" ,
\n" ); document.write( "the denominator, \"x%2B1\" , approaches zero.
\n" ); document.write( "As a consequence, the absolute value of the quotient \"f%28x%29=x%28x%2B5%29%2F%28x%2B1%29\" grows without bounds.
\n" ); document.write( "That makes \"highlight%28x=-1%29\" a \"highlight%28vertical%29\"\"highlight%28asymptote%29\" .
\n" ); document.write( "Also, for .
\n" ); document.write( "and \"f%28x%29-%28x%2B4%29=-4%2F%28x%2B1%29\" approaches zero as \"abs%28x%29\" increases.
\n" ); document.write( "That makes \"highlight%28y=4x%2B4%29\" an \"highlight%28asymptote%29\" .
\n" ); document.write( "With all that information, we can sketch the graph of \"f%28x%29\" ,
\n" ); document.write( "which looks like this \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "NOTE:
\n" ); document.write( "\"f%28x%29=%28x%5E3%2B4x%5E2-5x%29%2F%28x%5E2-1%29\" should be written as f(x)=(x^3+4x^2-5x)/(x^2-1),
\n" ); document.write( "because according with the algebra conventions known as \"order of operations\" rules,
\n" ); document.write( "multiplication and divisions are done before addition and subtraction, so
\n" ); document.write( "x^3+4x^2-5x/x^2-1 = \"x%5E3%2B4x%5E2-5x%2Fx%5E2-1\" .
\n" ); document.write( "The long horizontal line separating \"x%5E3%2B4x%5E2-5x\" from \"x%5E2-1\" is a very efficient grouping symbol:
\n" ); document.write( "it saves you from writing two pairs of parentheses.
\n" ); document.write( "However, when you have to key in symbols one at a time,
\n" ); document.write( "you have to key in all those parentheses that are implied.
\n" ); document.write( "Otherwise, you can get in trouble.
\n" ); document.write( "\"%280.735-0.004%29%2F%280.720-0.008%29=0.731%2F0.712=1.027\" (rounded to 3 decimal places).
\n" ); document.write( "Most calculators know about order of operations,
\n" ); document.write( "and if you key in
\n" ); document.write( "0.735 - 0.004 / 0.720 - 0.008 =\"0.735-0.004%2F0.720-0.008\" ,
\n" ); document.write( "you will get \"0.735-0.004%2F0.720-0.008=0.735-%220.00555...%22-0.008=0.735-%220.01355...%22=0.721\" (rounded to 3 decimal places). \n" ); document.write( "
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