Consider the following equation:

[tex]f(x)=\frac{x^2+4}{4 x^2-4 x-8}[/tex]

Name the vertical asymptote(s).

Find the zeros of the denominator 4 x 2 − 4 x − 8 by setting it to zero.
Simplify the equation 4 x 2 − 4 x − 8 = 0 to x 2 − x − 2 = 0 .
Factor the quadratic equation as ( x − 2 ) ( x + 1 ) = 0 , which gives x = 2 or x = − 1 .
Since the numerator x 2 + 4 is non-zero at these points, the vertical asymptotes are x = 2 , x = − 1 ​ .

Explanation

Problem Analysis We are given the function f ( x ) = 4 x 2 − 4 x − 8 x 2 + 4 ​ and asked to find its vertical asymptotes.

Finding Zeros of Denominator Vertical asymptotes occur where the denominator of a rational function is equal to zero and the numerator is non-zero. So, we need to find the zeros of the denominator 4 x 2 − 4 x − 8 .

Simplifying the Equation We set the denominator equal to zero: 4 x 2 − 4 x − 8 = 0 We can simplify this equation by dividing by 4: x 2 − x − 2 = 0

Factoring Quadratic Now, we factor the quadratic equation: ( x − 2 ) ( x + 1 ) = 0

Solving for x Solving for x , we get: x = 2 or x = − 1

Checking Numerator We need to check if the numerator is non-zero at these values. The numerator is x 2 + 4 . At x = 2 , the numerator is 2 2 + 4 = 4 + 4 = 8  = 0 At x = − 1 , the numerator is ( − 1 ) 2 + 4 = 1 + 4 = 5  = 0

Conclusion Since the numerator is non-zero at x = 2 and x = − 1 , the vertical asymptotes are x = 2 and x = − 1 .


Examples
Understanding vertical asymptotes is crucial in fields like physics and engineering, where functions often model real-world phenomena. For instance, in electrical engineering, the impedance of a circuit can be represented as a function of frequency. Vertical asymptotes in this function indicate resonant frequencies, where the circuit's behavior changes drastically. Similarly, in chemical engineering, reaction rates can be modeled by functions with asymptotes, representing conditions under which the reaction becomes unstable or uncontrollable. Identifying these asymptotes helps engineers design safer and more efficient systems.

Answered by GinnyAnswer | 2025-07-05