Let's examine this. The best you can do is to restate the function as:. If the degrees of the numerator and the denominator are the same, then the only division you can do is of the leading terms. If the degree is higher on top, then the division gives a polynomial whose degree is the difference between the degrees of the numerator and denominator. Since you'll only be doing rationals where the numerator's degree is at most 1 greater than the denominator's degree, then the division will only give you, at most, a linear straight-line expression.
In a sense, then, you're always using long division to find the horizontal or slant asymptote. Home Algebra Rational Functions. Still Confused?
Nope, got it. Play next lesson. Try reviewing these fundamentals first Polynomial long division Polynomial synthetic division. That's the last lesson Go to next topic.
Still don't get it? Review these basic concepts… Polynomial long division Polynomial synthetic division Nope, I got it. Play next lesson or Practice this topic. Play next lesson Practice this topic. Start now and get better math marks! Intro Lesson. Lesson: 1a. Lesson: 1b. The graph is symmetric about the y -axis if the function is even. The graph is symmetric about the origin if the function is odd.
Step 4: Find and plot any intercepts that exist. The x -intercept is where the graph crosses the x -axis. The y -intercept is where the graph crosses the y -axis. If you need a review on intercepts, feel free to go to Tutorial Equations of Lines. Step 5: Find and plot several other points on the graph.
Step 6: Draw curves through the points, approaching the asymptotes. Note that your graph can cross over a horizontal or oblique asymptote, but it can NEVER cross over a vertical asymptote. Example 7 : Sketch the graph of the function. This function cannot be reduced any further. This means that there will be no open holes on this graph. Vertical Asymptote: So now we want to find where the denominator is equal to Horizontal Asymptote: So now we want to compare the degrees of the numerator and the denominator.
If you said 0, you are correct. The leading term is the constant -1 and its degree is 0. Slant Asymptote: Since the degree of the numerator is NOT one degree higher than the degree of the denominator, there is not slant asymptote.
Since , the function is even. This means the graph is symmetric about the y -axis. You are correct if you said 0. This means there is NO x -intercept. This means there is NO y -intercept. So far we have not found any points to plot on the graph. Note how the vertical asymptote sections the graph into two parts.
Plugging in -1 for x we get: -1, Plugging in 1 for x we get: 1, Example 8 : Sketch the graph of the function. Since , the function is neither even nor odd.
Note how the vertical asymptotes section the graph into three parts. Plugging in 3 for x we get: 3, 1. Example 9 : Sketch the graph of the function.
The factor of x canceled out and there were no factors of x left in the denominator. Vertical Asymptote: So now we want to find where the new denominator is equal to Horizontal Asymptote: Since the degree of the numerator is one degree higher than the degree of the denominator, there is a slant asymptote and no horizontal asymptote.
Slant Asymptote: Applying long division to this problem we get: The answer to the long division would be. There are two x -intercepts: 4, 0 and -1, 0. Note that this will be open hole, as found in step 1. Plugging in -5 for x we get: -5, Plugging in -4 for x we get: -4, These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.
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