Final answer:
To graph f(x) = (x + 6)/(x + 2), find the vertical asymptote at x = -2 and the horizontal asymptote at y = 1. Plot the point (0,3) on the graph, and then draw the curve that approaches the asymptotes.
Step-by-step explanation:
The correct answer is option Mathematics. To graph the function f(x) = (x + 6)/(x + 2), start by identifying the vertical and horizontal asymptotes. A vertical asymptote occurs where the denominator of a fraction is zero (but not the numerator), which for this function is when x + 2 = 0, so at x = -2. This means there is a vertical asymptote at x = -2.
To find the horizontal asymptote, examine the degrees of the polynomials in the numerator and denominator. Since they are both first degree, the coefficients of x lead us to the horizontal asymptote y = 1. Now plot these asymptotes on your graph. At x very large in the positive or negative direction, the graph will approach the line y = 1.
For an additional point, choose a value for x and plug it into the function. For example, if x = 0, then f(0) = (0 + 6)/(0 + 2) = 3. So the point (0,3) lies on the graph. Plot this point and draw the graph with a curve approaching the asymptotes.
To find the horizontal asymptote, determine the limit of f(x) as x approaches positive or negative infinity. Since the degrees of the numerator and denominator are equal (both 1), the horizontal asymptote is y = the ratio of the leading coefficients, which is 1/1 or y = 1.
To further illustrate the function's behavior, we can plot an additional point on the graph. Let's choose x = 1. Plugging x = 1 into the function gives us f(1) = (1 + 6)/(1 + 2) = 7/3. So the point (1, 7/3) lies on the graph. Combining all this information, we can plot the graph of f(x) = (x + 6)/(x + 2) with the vertical asymptote x = -2, the horizontal asymptote y = 1, and the point (1, 7/3).