Consider the following equation:
[tex]f(x)=\frac{x^2+4}{4 x^2-4 x-8}[/tex]
Name the vertical asymptote(s).
Answer (1)
Factor the denominator of the rational function: 4 x 2 − 4 x − 8 = 4 ( x − 2 ) ( x + 1 ) .
Find the values of x that make the denominator zero: x = 2 and x = − 1 .
Verify that the numerator is not zero at these values: x 2 + 4 is non-zero at x = 2 and x = − 1 .
Conclude that the vertical asymptotes are x = 2 and x = − 1 : x = − 1 , x = 2 .
Explanation
Understanding Vertical Asymptotes We are given the function f ( x ) = 4 x 2 − 4 x − 8 x 2 + 4 and asked to find its vertical asymptotes. Vertical asymptotes occur at values of x where the denominator is zero and the numerator is non-zero.
Factoring the Denominator First, we need to factor the denominator of the function: 4 x 2 − 4 x − 8 = 4 ( x 2 − x − 2 ) = 4 ( x − 2 ) ( x + 1 ) So the denominator is 4 ( x − 2 ) ( x + 1 ) .
Finding Potential Asymptotes Next, we set the denominator equal to zero and solve for x :
4 ( x − 2 ) ( x + 1 ) = 0 This gives us x = 2 or x = − 1 .
Checking the Numerator Now, we need to check if the numerator is non-zero at these values of x . The numerator is x 2 + 4 .
If x = 2 , the numerator is 2 2 + 4 = 4 + 4 = 8 , which is not zero. If x = − 1 , the numerator is ( − 1 ) 2 + 4 = 1 + 4 = 5 , which is not zero.
Conclusion Since the denominator is zero at x = 2 and x = − 1 , and the numerator is non-zero at these points, the vertical asymptotes are x = 2 and x = − 1 .
Examples
Vertical asymptotes are useful in various real-world applications. For example, in physics, they can represent the behavior of a system approaching a singularity, such as the electric field near a point charge. In economics, they can model situations where a small change in one variable leads to a drastic change in another, like the cost of production approaching infinity as output approaches capacity. Understanding vertical asymptotes helps us analyze and predict the behavior of functions in extreme conditions.
