Graph the function y = 1/(x - 2) and state the asymptotes

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Let's graph the function y equals 1 divided by x minus 2. First, we need to identify the asymptotes. Since the denominator equals zero when x equals 2, we have a vertical asymptote at x equals 2. As x approaches infinity, the function approaches zero, giving us a horizontal asymptote at y equals 0. The graph consists of two branches: one to the left of x equals 2 where the function is negative, and one to the right where the function is positive. Now let's analyze the behavior of this function in more detail. As x approaches 2 from the left, y approaches negative infinity. As x approaches 2 from the right, y approaches positive infinity. As x approaches either positive or negative infinity, y approaches zero, which is our horizontal asymptote. The function has no x-intercepts because the equation 1 divided by x minus 2 equals 0 has no solution. However, it does have a y-intercept at the point (0, -1/2), which we can find by substituting x equals 0 into the function. Let's understand how our function y equals 1 divided by x minus 2 relates to the parent function y equals 1 divided by x. The parent function has a vertical asymptote at x equals 0 and a horizontal asymptote at y equals 0. Our function is a horizontal transformation of the parent function - specifically, a shift 2 units to the right. This transformation moves the vertical asymptote from x equals 0 to x equals 2, and shifts all points of the graph 2 units to the right. Notice that the horizontal asymptote remains at y equals 0, as horizontal shifts don't affect horizontal asymptotes. To better understand the function y equals 1 divided by x minus 2, let's calculate some specific points on the graph. For x equals 0, y equals 1 divided by 0 minus 2, which is 1 divided by negative 2, or negative one-half. For x equals 1, y equals 1 divided by 1 minus 2, which is 1 divided by negative 1, or negative 1. For x equals 3, y equals 1 divided by 3 minus 2, which is 1 divided by 1, or simply 1. For x equals 4, y equals 1 divided by 4 minus 2, which is 1 divided by 2, or 0.5. And for x equals 6, y equals 1 divided by 6 minus 2, which is 1 divided by 4, or 0.25. Plotting these points helps us confirm the shape of our graph. Let's summarize what we've learned about the function y equals 1 divided by x minus 2. This is a rational function with a vertical asymptote at x equals 2, where the denominator equals zero, and a horizontal asymptote at y equals 0, which the function approaches as x approaches infinity. The graph consists of two separate branches: one in the region where x is less than 2, where the function takes negative values, and another in the region where x is greater than 2, where the function takes positive values. As x approaches 2 from the left, y approaches negative infinity, and as x approaches 2 from the right, y approaches positive infinity. This function is a horizontal transformation of the parent function y equals 1 divided by x, shifted 2 units to the right.