Algebra: Conic sections - ellipse, parabola, hyperbola Section Solvers Solvers Lessons Lessons Answers archive Answers

Click here to see ALL problems on Quadratic-relations-and-conic-sections Question 124956: Find the equations of the horizontal and vertical asymptotes for the following. Tye none if the function does not have an asymptote.

a. f(x)=2x+3/x+2

b. g(x)=5x/x^2+1

Answer by jim_thompson5910(35256) (Show Source):





a)



Start with the given function







Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .





Horizontal Asymptote:

Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:



To find the horizontal aysmptote, first we need to find the leading coefficients of the numerator and the denominator.



Looking at the numerator , the leading coefficient is



Looking at the denominator , the leading coefficient is



So the horizontal aysmptote is the ratio of the leading coefficients. In other words, simply divide by to get





So the horizontal asymptote is











--------------------------------------------------







Vertical Asymptote:

To find the vertical aysmptote, just set the denominator equal to zero and solve for x



Set the denominator equal to zero





Subtract 2 from both sides





Combine like terms on the right side





So the vertical asymptote is





Notice if we graph , we can visually verify our answers:



Graph of with the horizontal asymptote (blue line) and the vertical asymptote (green line) I'll do the first one to get you starteda)Start with the given functionLooking at the numerator, we can see that the degree issince the highest exponent of the numerator is. For the denominator, we can see that the degree issince the highest exponent of the denominator isSince the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:To find the horizontal aysmptote, first we need to find the leading coefficients of the numerator and the denominator.Looking at the numerator, the leading coefficient isLooking at the denominator, the leading coefficient isSo the horizontal aysmptote is the ratio of the leading coefficients. In other words, simply dividebyto getSo the horizontal asymptote is--------------------------------------------------To find the vertical aysmptote, just set the denominator equal to zero and solve for xSet the denominator equal to zeroSubtract 2 from both sidesCombine like terms on the right sideSo the vertical asymptote isNotice if we graph, we can visually verify our answers:Graph ofwith the horizontal asymptote(blue line) and the vertical asymptote(green line) You can put this solution on YOUR website!

