Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(t)=\frac{t^{2}}{t^{2}-9} $$

Step-by-Step Solution

Answer

The function \(f(t)=\frac{t^{2}}{t^{2}-9}\) has vertical asymptotes at \(t = 3\) and \(t = -3\), and a horizontal asymptote at \(y = 1\).

Step 1: Identify the value(s) of t that make the denominator to be zero

To find the value(s) of t for which the function's denominator is zero, we need to solve the following equation: $$ t^2-9=0 $$

Step 2: Solve the equation to find the vertical asymptotes

We have the quadratic equation \(t^2 - 9 = 0\). This equation can be factored as \((t-3)(t+3) = 0\). This equation has two solutions: \(t=3\) and \(t=-3\). These are the values of t where the denominator becomes zero. Hence, we have: - Vertical asymptote 1: \(t = 3\) - Vertical asymptote 2: \(t = -3\) Now, let's move on to find the horizontal asymptote.

Step 3: Determine the horizontal asymptote

We can find the horizontal asymptote by finding the limit of the function as t approaches infinity or negative infinity. The function is given as: $$ f(t)=\frac{t^{2}}{t^{2}-9} $$ We can determine the behavior of the function for large positive and negative values of t. Let's find the limit as t approaches infinity: $$ \lim_{t\rightarrow\infty}{\frac{t^{2}}{t^{2}-9}} $$ In this case, the degree of the numerator and the denominator are equal (both are 2). To find the limit in this case, we can simply take the ratio of the leading coefficients, which are 1 in both cases: $$ \lim_{t\rightarrow\infty}{\frac{t^{2}}{t^{2}-9}} = \frac{1}{1} = 1 $$ Since the limit is a constant value when t approaches infinity, we can conclude that there is a horizontal asymptote at \(y = 1\). #Results#: - Vertical asymptotes: \(t = 3\) and \(t = -3\) - Horizontal asymptote: \(y = 1\)

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function's denominator equals zero because the function becomes undefined at these points. For the given function, \(f(t) = \frac{t^2}{t^2 - 9}\), finding vertical asymptotes involves solving \(t^2 - 9 = 0\). By factoring, we have \((t - 3)(t + 3) = 0\), meaning \(t = 3\) and \(t = -3\) are solutions.

• When \(t = 3\) or \(t = -3\), the denominator of the function becomes zero, thus the function shoots off to infinity or negative infinity.
• This creates vertical asymptotes, which are essentially lines t = 3 and t = -3 that the graph approaches but never touches.

Understanding vertical asymptotes helps in sketching rational function graphs, identifying points of discontinuity, and realizing where the function isn't defined.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a graph as the input value t (in this function) becomes very large or very small. For our function \(f(t) = \frac{t^2}{t^2 - 9}\), both the numerator and the denominator are quadratic terms.

To find the horizontal asymptote, note that the degrees (highest exponent) of the numerator and denominator are the same (both 2 in this case). This implies that as \(t\) approaches infinity, the fractions' leading coefficients determine the horizontal asymptote, which is a constant value:

• Leading coefficients for both numerator and denominator are 1, so we have \(\frac{1}{1} = 1\).
• The horizontal asymptote, therefore, is \(y = 1\).
  
  

The horizontal asymptote tells us that as \(t\) becomes extremely large or small, the values of \(f(t)\) approach 1, but the graph never truly meets this line.

Rational Functions

Rational functions are ratios of two polynomial expressions, where one polynomial is divided by another. In the function \(f(t) = \frac{t^2}{t^2 - 9}\), \(t^2\) is the numerator and \(t^2 - 9\) is the denominator indicating it's a rational function.

These functions often present interesting and predictable properties such as:

• Vertical asymptotes: Locations where the function is undefined due to division by zero, as is seen when \(t = 3\) and \(t = -3\).
• Horizontal asymptotes: Describe how the function behaves at extreme positive or negative values of the input. In our case, the grip tightens towards \(y = 1\).
• Holes: Occur when a factor of the denominator cancels out with a factor in the numerator, which isn't present in this specific function but noteworthy in understanding other rational functions.

Studying rational functions helps predict long-term behavior of graphs and locate discontinuities that are critical in mathematical modeling and analysis.