Find the vertical asymptotes of the rational function.
[tex]h(x)=\frac{-3 x^2}{x^2+5 x-14}[/tex]
Answer (1)
Set the denominator of the rational function equal to zero: x 2 + 5 x − 14 = 0 .
Factor the quadratic equation: ( x + 7 ) ( x − 2 ) = 0 .
Solve for x to find potential vertical asymptotes: x = − 7 and x = 2 .
Verify that the numerator is not zero at these x values. Since − 3 x 2 is not zero at x = − 7 and x = 2 , the vertical asymptotes are x = − 7 , x = 2 .
Explanation
Understanding the problem We are asked to find the vertical asymptotes of the rational function h ( x ) = x 2 + 5 x − 14 − 3 x 2 . Vertical asymptotes occur where the denominator of a rational function is equal to zero and the numerator is not zero at the same point.
Setting the denominator to zero To find the vertical asymptotes, we need to find the values of x for which the denominator is zero. So, we set the denominator equal to zero: x 2 + 5 x − 14 = 0
Factoring the quadratic equation Now, we solve the quadratic equation x 2 + 5 x − 14 = 0 . We can solve this by factoring. We are looking for two numbers that multiply to -14 and add to 5. These numbers are 7 and -2. So, we can factor the quadratic as follows: ( x + 7 ) ( x − 2 ) = 0
Finding the roots of the denominator Setting each factor equal to zero gives us the solutions: x + 7 = 0 ⇒ x = − 7 x − 2 = 0 ⇒ x = 2
Checking the numerator Now we need to check if the numerator is zero at these points. The numerator is − 3 x 2 .
At x = − 7 , the numerator is − 3 ( − 7 ) 2 = − 3 ( 49 ) = − 147 = 0 .
At x = 2 , the numerator is − 3 ( 2 ) 2 = − 3 ( 4 ) = − 12 = 0 .
Conclusion Since the numerator is not zero at x = − 7 and x = 2 , the vertical asymptotes are x = − 7 and x = 2 .
Examples
Understanding vertical asymptotes is crucial in fields like physics and engineering, where rational functions model various phenomena. For instance, in circuit analysis, the transfer function of a circuit can be a rational function, and its vertical asymptotes indicate resonant frequencies where the circuit's behavior changes drastically. Similarly, in fluid dynamics, rational functions can describe flow rates, and their asymptotes can represent critical points where the flow becomes unstable or undefined. Identifying these asymptotes helps engineers design stable and efficient systems, avoiding potentially hazardous conditions.
