Math Stuart Jones. 4.3 Curve Sketching

Transcription

1 4.3 Curve Sketching

2 In this section, we combine much of what we have talked about with derivatives thus far to draw the graphs of functions. This is useful in many situations to visualize properties of the function. The first idea we will approach is that of a vertical asymptote. An asymptote is a line that the graph approaches but never crosses. Vertical Asymptotes The vertical line x = a is a vertical asymptote of the graph of the function of f if lim f (x) = or x a + OR lim f (x) = or x a

3 Finding Vertical Asymptotes If f (x) is a rational function, f (x) = P(x) Q(x), where P and Q are polynomials, then f (x) has vertical asmyptotes everywhere Q(x) = 0, so long as P(x) 0 at those points.

4 Find the vertical asymptotes (if any) of f (x) = x2 4 x 2 +5x+6

5 Find the vertical asymptotes (if any) of f (x) = Solution: Since this is a rational function, we have vertical asymptotes everywhere my denominator is 0, as long as my numerator is not 0 in those places. So we set the denominator equal to 0 and solve: x2 4 x 2 +5x+6 x 2 + 5x + 6 = 0 = (x + 2)(x + 3) = 0 = x = 2, x = 3 For my numerator: x 2 4 = 0 = (x + 2)(x 2) = 0 = x = 2, x = 2 So x=2 is NOT a vertical asymptote, since my numerator and denominator have this in common (we can cancel that factor). Hence, my only vertical asymptote is x=-3.

6 The graph of the previous example is shown here. Notice the vertical asymptote at x=

7 Horizontal Asymptotes The line y = b is a horizontal asymptote of the graph of a function f (x) is either lim f (x) = b OR lim x f (x) = b x NOTE: Polynomials never have any asymptotes! So, if you have a polynomial, you can know immediately there are no horizontal or vertical asymptotes.

8 Let s use our previous example. Find the horizontal asymptotes (if any) of f (x) = x2 4 x 2 +5x+6.

9 Let s use our previous example. Find the horizontal asymptotes (if any) of f (x) = x2 4. HINT: The answer is y=. Can x 2 +5x+6 you show this?

10 Find any horizontal or vertical asymptotes of the function f (x) = x x 2 6x+8

11 Find any horizontal or vertical asymptotes of the function f (x) = x x 2 4

12 Find any horizontal or vertical asymptotes of the function f (x) = x x 2 4 I work this one out since it is an odd one. If we simply plug in infinity, we get infinity/infinity, which is indeterminate form. So... We look at the highest power in the denominator (x 2 ), and take the square root of it, and flip it, x. We then multiply top and bottom by that. (You may just want to memorize this trick.) Now: x x x x 2 4 = (x x 2 4) 2 (Continued.../4) lim x = 4 x 2 = = x 2 4 x 2 x 2 = = 4 x 2

13 So, we have a horizontal asymptote at y=. We also have to check when x goes to negative infinity, just in case. We cannot take the x and put it into the denominator since it is negative. So we have to factor out a negative before we put it into the square root. x x x x 2 4 = So, when we take the limit: x 2 x 2 4 x 2 = 4 x 2 lim x = 4 = x 2 = So, y=- is also a horizontal asymptote. (Continued...2/4)

14 Finally, we need to check for vertical asymptotes: x 2 4 = 0 x 2 4 = 0 = (x + 2)(x 2) = 0 And our numerator is only 0 when x=0, so our two vertical asymptotes are x=-2 and x=2. The graph of this function is shown on the next page. Notice the horizontal asymptotes at y= and y=- and the horizontal asymptotes at x=2 and x=-2. (Continued...3/4)

15 Here is the graph of f (x) = x x

16 We now combine everything from the previous sections. We can use this to sketch (roughly draw) the graph of a function. Curve Sketching Steps Determine domain of f. Where is f undefined? 2 Find x and y intercepts, if reasonable to do so. 3 Determine end behavior of f. 4 Determine intervals where f is increasing and decreasing. 5 Find the critical points - relative min, relative max, inflection points, and undefined points, as well as concavity. 6 If needed, plot a few other test points to finalize the sketch as needed. This graph will not necessarily be perfect, but only a sketch.

17 Sketch the graph of f (x) = 3x 3 x 2 + 4x.

18 Sketch the graph of f (x) = 3x 3 x 2 + 4x. Here is the graph:

19 Sketch the graph of f (x) = x x+

20 Sketch the graph of f (x) = x x+ Here is the graph:

21 Sketch the graph of f (x) = x 4 6x 2.

22 Sketch the graph of f (x) = x 4 6x 2. Here is the graph:

23 The Bottom Line Vertical asymptotes are vertical lines (x=number) where the function is undefined. In this case, y goes to infinity (or -infinity) as x approaches this number. Horizontal asymptotes are horizontal lines (y=number) that the function approaches as x goes to infinity (or -infinity) Polynomials have no asymptotes. You can use all of the information from this and previous sections to roughly draw graphs of many functions. You will need to practice this!