1. Some rational functions have asymptotes, others have holes, and some have both. Explain how you can identify, without graphing, which graphical features a rational function will have.
Can someone explain this thoroughly ? I don't understand. thanks in advance.
the culprit for asymptotes and holes is the denominator.
If for some value of the variable, then denominator is zero, but the numerator is NOT zero, you will have an asymptote.
If for some value of the variable, then denominator is zero, but the numerator is ALSO zero, you will have a hole
e.g. y = (x-2)/(x^2 - 4)
notice this reduces to y = 1/(x+2)
so if x = 2 in the original we get 0/0, so there is a hole at (2,1/4)
if x = -2 we get -4/0 in the original, so x = -2 is an asymptote
how would we know if it has both holes and asymptotes?
What is the domain and range of 2x^2-18/x^2+3x-10
To identify the graphical features of a rational function without graphing it, you can follow these steps:
1. Analyze the numerator and denominator: Start by examining the rational function's numerator and denominator. Determine the degrees of each polynomial.
- If the degree of the numerator is greater than the degree of the denominator, there will be a slant (oblique) asymptote.
- If the degree of the numerator is equal to the degree of the denominator, there may be a horizontal asymptote.
- If the degrees are different, there will be a vertical asymptote.
2. Find vertical asymptotes: Set the denominator equal to zero and solve for the variable. The values that make the denominator zero will give the vertical asymptotes of the function. These vertical lines act as bounds for the graph.
3. Determine horizontal asymptotes: Depending on the degrees of the numerator and denominator, there are three different possibilities:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
- If the degrees are the same, divide each term of the numerator by the highest power of the denominator. The resulting quotient represents the horizontal asymptote.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there may be a slant asymptote.
4. Identify holes (removable discontinuities): Factor both the numerator and the denominator, then cancel out any common factors. The values that make the denominator zero and do not cancel out with the numerator will result in holes in the graph.
By following these steps, you can determine the presence of vertical asymptotes, horizontal asymptotes, and holes with a rational function, without needing to graph it.