f(x) = (2x² + 1)/(x² - 4). Identify the horizontal asymptote and both vertical asymptotes.

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Let's analyze the rational function f(x) equals 2x squared plus 1, divided by x squared minus 4. First, we need to identify the vertical asymptotes. These occur where the denominator equals zero. Setting x squared minus 4 equal to zero and factoring, we get (x minus 2) times (x plus 2) equals zero. Solving this equation gives us x equals 2 or x equals negative 2. These are our vertical asymptotes, shown as red dashed lines on the graph. Now let's find the horizontal asymptote of this function. To do this, we compare the degrees of the numerator and denominator. Both the numerator 2x squared plus 1 and the denominator x squared minus 4 have degree 2. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y equals 2, shown as the green dashed line on the graph. Notice how the function approaches this horizontal asymptote as x approaches positive or negative infinity. Let's examine the behavior of our function near the asymptotes. Near the vertical asymptotes at x equals 2 and x equals negative 2, the function approaches positive or negative infinity. As x approaches 2 from the left, the function approaches negative infinity, while from the right, it approaches positive infinity. For x approaching negative 2, the behavior is reversed. For the horizontal asymptote, as x approaches positive or negative infinity, the function approaches y equals 2. We can also evaluate specific points, such as at x equals 0, where the function equals negative one-fourth or negative 0.25, as shown by the yellow dot on the graph. These behaviors help us understand how the function behaves throughout its domain. Now, let's verify our asymptotes algebraically. For the horizontal asymptote, we examine the limit of the function as x approaches infinity. Dividing both numerator and denominator by x squared, we get the limit of 2 plus 1 over x squared, divided by 1 minus 4 over x squared. As x approaches infinity, the terms with x squared in the denominator approach zero, leaving us with 2 divided by 1, which equals 2. This confirms our horizontal asymptote at y equals 2. For the vertical asymptotes, we substitute x equals 2 and x equals negative 2 into the function. In both cases, the denominator becomes zero while the numerator is non-zero, confirming that these are indeed vertical asymptotes. Therefore, our function has a horizontal asymptote at y equals 2 and vertical asymptotes at x equals 2 and x equals negative 2. To summarize our analysis of the rational function f(x) equals 2x squared plus 1, divided by x squared minus 4: We identified two vertical asymptotes at x equals 2 and x equals negative 2, which occur where the denominator equals zero. We also found a horizontal asymptote at y equals 2, determined by the ratio of the leading coefficients in the numerator and denominator. Near the vertical asymptotes, the function approaches positive or negative infinity, while as x approaches infinity, the function approaches the horizontal asymptote. Understanding asymptotes is crucial for graphing rational functions and analyzing their behavior. These concepts apply to many real-world scenarios in science, engineering, and economics where rational functions model important relationships.