This is a more advanced topic that shows up sometimes once a test, often in the 40-60 range. See the difficulty range for more info. If your ACT® Math score isn't consistently 25+, I wouldn't worry about this topic yet.
The test is likely to only ask you one question about asymptotes. This topic is one of those things that you either know or don't. The mathematical application tends not to be anything more than showing you understand the basic rules of asymptotes.
Asymptotes are important in understanding the behavior of functions, especially when dealing with limits. They help us predict how graphs behave as they extend towards very large numbers, which is useful in fields like physics and engineering when modeling real-world phenomena.
What is an asymptote?
An asymptote is a line where the distance between the curve and the line gets closer and closer to zero as one of the x or y coordinates (or both) become extremely large (approaching infinity).
Horizontal asymptotes are parallel to the x-axis, whereas vertical asymptotes are perpendicular. Oblique asymptotes happen once in a blue moon on the ACT® Math section. It's not worth most students' time to cover them here.
If these definitions don't quite make sense yet, reference them as you see the examples below.
Asymptote Rules
Horizontal Asymptote Rules:
- Numerator Degree > Denominator Degree: NoneExplanation: The function will grow without bound, so there's no horizontal line it approaches.
- Numerator Degree = Denominator Degree: leading coefficient of numerator / leading coefficient of denominatorExplanation: The function approaches this ratio as x becomes very large.
- Numerator Degree < Denominator Degree: Explanation: The denominator grows faster than the numerator, so the fraction approaches zero.
Vertical Asymptote Rules:
Set the denominator to zero.
Consider these examples
Question 1:
Find the vertical and Horizontal Asymptotes of
Click for the Answer
Vertical Asymptote:
Vertical asymptotes occur where the denominator is zero and the numerator is not zero.
Set the denominator equal to zero and solve for x:
Therefore, there is a vertical asymptote at
Horizontal Asymptote:
When the degrees of the leading coefficients (x) are the same, then you can divide them to find the horizontal asymptote:
Question 2:
Find the Horizontal Asymptote of
Click for the Answer
Horizontal Asymptote:
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:
Question 3:
Find the horizontal Asymptote of
Click for the Answer
Horizontal Asymptote:
Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Conclusion
Consider a company's market share. As a successful company grows, it might capture more and more of the market, but it can never quite reach 100% market share. The graph of the company's market share over time might approach 100% (which acts like a horizontal asymptote) but never quite reach it.
That's how you can think about asymptotes. They represent a fucntion that never reaches infinity