On-line Math 21

4.5  Curve Sketching

Example 6

f(x) = 2x+3 x+1

Solution

   Asymptotes

Here it is easy to see, since the limit at -1 is infinte, that x = -1 is a vertical asymptote. Also, y = 2 is a horizontal asymptote, and these are clearly the only possibile asymptotes.

Now, to get the tails of the graph, as indicated, you have to look more closely. For x large, the numerator is 1 more than twice the denominator, so f(x) will be a bit more than 2 for x >> 0 . But on the other hand, for x << 0 (that means `` x much less than 0''), the denominator is also 1 more than twice the denominator. However, this time, both the numerator and denominator are negative, so 1 more than twice the denominator has absolute value less than twice the absolute value of the denominator. So, the fraction is a bit less than 2 for x << 0 .

For the vertical asymptote, since for x > -1 , but x near -1 , f(x) is positive, since both numerator (near +1) and denominator (near 0, but positive) are positive. But it is becoming infinite as x approaches 0 , so it must be that

lim x® -1+  2x+3 x+1 = +¥,

so the curve approaches the asymptote at the top on the right side. Similarly,

lim x® -1-  2x+3 x+1 = -¥,

so the curve approaches the asymptote at the bottom on the left side.

   Intercepts

f(0) = 3 , so (0,3) is the y -intercept. There is only one x -intercept, where the numerator is 0 , at x = -3/2 , so (-3/2,0) is the x- intercept.

   Increasing/decreasing, and critical points

f¢(x) = 2(x+1)-1(2x+3) (x+1)2 = -1 (x+1)2 ,

so the curve is always decreasing. No critical points.

   Concavity, and inflection points

f¢¢(x) = æ ç è -1 (x+1)2 ö ÷ ø ¢ = (-1)(-2)(x+1)-3 = 2 (x+1)3 ,

so the curve is concave up for x > -1 and concave down for x < -1 .

   Draw the graph

Use this information to draw a fair representation of the graph.

• The first information you plot are the asymptotes and ``tails'' [Link to ex6-1.gif]
• Then find intercepts and endpoints; [Link to ex6-2.gif]
• Then plot the critical points (and values). Of course, there are none here.
• Plot the inflection points. None of these in this example, either.
• Then fill in the graph, connecting the dots and making sure that the graph you draw is increasing/decreasing where the number line indicates it should be, and concave-up or concave-down where the sign of the second derivative indicates. [Link to ex6-3.gif]

[Each of these items should trigger the appearance of a new drawing with that information added.]

Copyright (c) 2000 by David L. Johnson.