1 / 11

2. Solve for f(x) = k where k is the y-value of the asymptote.

From the previous section, we know the graph of the function can never cross the vertical asymptotes. However, this is not true for some functions with horizontal asymptotes. Below are examples of some functions whose graph does intersect the horizontal asymptote.

cadee
Download Presentation

2. Solve for f(x) = k where k is the y-value of the asymptote.

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. From the previous section, we know the graph of the function can never cross the vertical asymptotes. However, this is not true for some functions with horizontal asymptotes. Below are examples of some functions whose graph does intersect the horizontal asymptote. Procedure: To find points of intersection of the graph and the horizontal asymptote. To obtain an accurate drawing of the function, we need to find values (if any) where the graph intersects the horizontal asymptote. If this point(s) exists, the coordinate has a y value the same as the horizontal asymptote. 1. Find the horizontal asymptote. 2. Solve for f(x) = k where k is the y-value of the asymptote. 3. These solutions are the x-values of the intersection. The y-value is the y-value of the asymptote.

  2. Example 1. Find the intersection of the horizontal asymptote and the function. (Do not graph.) Solution: Your Turn Problem #1 Find the intersection of the horizontal asymptote and the function 1. Find the horizontal asymptote. 2. Set the function equal to y = 3 and solve. Answer: The graph intersects the horizontal asymptote at (–2, 3).

  3. We will now discuss the case where the degree of the numerator is exactly one more than the degree of the denominator. In this case, there is no horizontal asymptote. We have what is called an oblique asymptote. Here are some functions where the degree of the numerator is one more than the degree of the denominator. Therefore, these functions have an oblique asymptote and not a horizontal asymptote. • Recall horizontal asymptotes: • If the degree of the numerator < degree of the denominator, y = 0 is the horizontal asymptote. • If the degree of the numerator = degree of the denominator, y = a/b where a and b are the leading coefficients is the horizontal asymptote. Note: We will not cover rational functions where the degree of the numerator is greater than the denominator by more than 1. This concept is beyond the scope of this course. Next Slide

  4. Oblique Asymptotes An oblique asymptote is a line that is neither horizontal nor vertical. Here are some graphs with an oblique asymptote. Compared to finding horizontal and vertical asymptotes, obtaining the oblique asymptote is a little more complicated. The asymptote is a line that is neither horizontal nor vertical. Therefore it is of the form: y = ax + b where m  0. Procedure: To find an oblique asymptote if the degree of the numerator is one more than the degree of the denominator. 1. Divide the denominator into the numerator. Long division or synthetic division may be used. 3. We are concerned with values as x gets infinitely large or infinitely small. As this occurs, the fractional remainder approaches zero. Therefore, the graph of the function approaches the line y = ax + b. This is the oblique asymptote.

  5. Example 2. Find the oblique asymptote of the function -2 -1 1 4 2 0 The graph oblique asymptote is y = 2x + 1. Answer: Your Turn Problem #2 Find the oblique asymptote of the function 1. Divide denominator into numerator. 2. Write out the quotient with the remainder. • The oblique asymptote is y = ax + b. (Simply disregard the remainder) Answer: y = 3x – 11

  6. Points of intersection of the graph and the oblique asymptote. There are functions where the graph may intersect the oblique asymptote. Below is an example of this intersection. Procedure: To find points of intersection of the graph and the oblique asymptote. The procedure for finding this intersection is the same as finding the point of intersection of the graph a horizontal asymptote. Here we will show the procedure for finding the intersection with an example. However, there will be no your turn problems, homework problems or exam questions covering this topic. 1. Find the oblique asymptote. 2. Solve for f(x) = ax+b. 3. If there is a solution, this is the x-values of the intersection. 4. Solve for the y-value using the solution found and the the equation of the oblique asymptote.

  7. Since y = x – 2, Example: Find the intersection of the oblique asymptote and the function: Solution: 1. Find the oblique asymptote by dividing the denominator into the numerator. The line y = x – 2 is the oblique asymptote 2. Set the function equal to x – 2 and solve. The following slide is a general strategy for graphing rational functions. Not all instructors will require all steps to be followed. Notice that finding the intersection of an oblique asymptote and the function may be somewhat complicated.

  8. General Strategy for Graphing a Rational Function y = x + 2 x=2 2 4 y=x + 2 x = 2 1 2 5 x x y -1 -2/3 1 -2 3 10 4 8 ½ Next Slide 1. Find the vertical asymptotes by setting the denominator equal to zero. 2. Find the horizontal or oblique asymptote. There can be only one non-vertical asymptote. 3. Determine if the graph will intersect the horizontal asymptote by solving for f(x) = k, where k is the y-value of the horizontal asymptote. 4. Plot a enough points in each interval to obtain an accurate graph. 5. Draw in the graph showing the function approaching the asymptotes. 1. Find vertical asymptote. 2. Find oblique asymptote. 3. There is no H.A. for the graph to intersect. 4. Find points. 5. Draw in graph.

  9. Your Turn Problem #3 x=-3 f(x) x = –3 vertical asymptote: y=x –3 oblique asymptote: y = x – 3 x x y x y -4 -17 -5 -13 -6 -12 1/3 -2 5 -1 1 1 1/2 2 1 3 1 2/3

  10. x y -3 30 3 12/49 4 15/32 5 2/3 -5 84 -10 9 -9 10 14/25 -1 0 2 0 V.A. x= -4 (set denominator = 0 and solve) H.A. y = 3 f(x) x=-3 y=3 Yes: The intersection is at (-2,3). This means the graph will go through the asymptote, then come back and approach the horizontal asymptote. x Plot enough points and graph. Find the asymptotes. Does graph intersect H.A.?

  11. Your Turn Problem #4 f(x) H.A. y = 0 x y (1,2) 1 2 2 1 3/5 3 1 1/5 4 16/17 10 40/101 -1 -2 -2 -1 3/5 -3 -1 1/5 -4 -16/17 -10 -40/101 x (-1,-2) There may be functions whose graph is a little different from the examples in this lesson. Keep in mind the basic procedure. Check for symmetry. Find all asymptotes, intercepts, and intersections with the horizontal asymptote. Plot enough points to obtain an accurate sketch. The End B.R. 3-05-07

More Related