Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ h(s)=\frac{3 s+4}{s^{2}-16} $$

Short answer

The vertical asymptotes of the function \( h(s) = \frac{3s+4}{s^2-16} \) are \(s = 4\) and \(s = -4\).

Step-by-step solution

Set the Denominator Equal to Zero

To find the possible points of the vertical asymptotes, the denominator \(s^2 - 16\) needs to be set equal to zero, and then solved for \(s\). In other words, solve the following equation: \(s^2 - 16 = 0 \).

Solve the Equation

The above equation is quadratic and can be factored following the difference of squares rule, \(a^2 - b^2 = (a - b)(a + b)\). Factoring gives \((s-4)(s+4)=0\). The solutions are \(s=4\) and \(s=-4\). These are potentially the vertical asymptotes.

Check the Numerator

Lastly, check if these solutions also make the numerator zero. If they do, then these are not the vertical asymptotes. Substitute into \(3s + 4\), and it can be seen that neither \(s = 4\) nor \(s = -4\) makes the numerator zero. So, the original function indeed has the vertical asymptotes at \(s = 4\) and \(s = -4\).

Key concepts

Vertical Asymptotes in Calculus

In calculus, vertical asymptotes are lines that a graph approaches but never actually reaches. These occur in functions that have certain points where they cannot be defined. Typically, finding vertical asymptotes involves looking at the points where the function goes to infinity. For rational functions, this usually happens where the denominator of the function is zero and the numerator is not.

To find vertical asymptotes, you’ll first want to identify the zeros of the denominator, as these are your potential asymptotes. From the exercise provided, the denominator is set to zero, and the quadratic equation is solved. If the numerator does not become zero at these points, then they are indeed vertical asymptotes. It’s a process of elimination to ensure the function doesn’t simplify to remove these points of undefined value.

Rational Functions

Rational functions are fractions that involve polynomials. They are written in the form (p(x))/(q(x)), where both (p(x)) and (q(x)) are polynomials and (q(x)) is not equal to zero. They can model many real-world scenarios, such as rates of change. The key feature of rational functions related to calculus is their behavior near points that lead the denominator towards zero, which often results in vertical asymptotes.

When examining such functions, it’s important to simplify them if possible to identify asymptotes correctly. In the case of the exercise given, simplification is not necessary, and upon setting the denominator to zero, the potential vertical asymptotes are identified.

Difference of Squares

The difference of squares is a pattern in algebra that describes how two squared terms separated by a subtraction sign can be factored. The general form is (a^2 - b^2) which equals (a - b)(a + b). It's an important concept to master as it often simplifies the process of solving quadratic equations, such as finding zeros of polynomials and potential vertical asymptotes for rational functions.

Understanding this pattern allows for quick factorization. As demonstrated in the exercise, the denominator (s^2 - 16) is recognized as a difference of squares since it can be written as (s^2 - 4^2) and factored into (s - 4)(s + 4). This crucial step can sometimes reveal solutions that weren't apparent at first glance.

Quadratic Equations

Quadratic equations form the backbone of many algebraic processes and are equations of the second degree, usually in the form of (ax^2 + bx + c = 0). These equations are recognizable by their characteristic parabolic graphs.

To solve a quadratic equation, you can factorize, complete the square, use the quadratic formula, or graphically find the x-intercepts. In the context of finding vertical asymptotes for rational functions, solving the quadratic equations that result from setting the denominator to zero is essential. As in the exercise, after factoring using the difference of squares, the solutions of the quadratic equation indicate potential points where vertical asymptotes might occur.