Final answer:
To sketch the graph of the function f(x) = (x^2 - x - 20)/(x + 4), we can identify the vertical asymptote, horizontal asymptote (if it exists), and the x and y intercepts. There is a vertical asymptote at x = -4, no horizontal asymptote, x-intercepts at x = 5 and x = -4, and a y-intercept at y = -5.
Step-by-step explanation:
The given function is f(x) = (x^2 - x - 20)/(x + 4). To sketch the graph, we can first identify the vertical asymptotes by finding the values of x for which the denominator becomes zero. In this case, x = -4 is the vertical asymptote since it makes the denominator zero. Next, we find the horizontal asymptote by looking at the highest power of x in the numerator and denominator. Here, the degree of the numerator is 2 and the degree of the denominator is 1, so there is no horizontal asymptote.
We can also find the x-intercept by setting the numerator equal to zero and solving for x. x^2 - x - 20 = 0 can be factored as (x - 5)(x + 4) = 0, giving us x = 5 or x = -4. Therefore, the graph intersects the x-axis at x = 5 and x = -4. Additionally, we can find the y-intercept by plugging in x = 0 into the function. f(0) = (-20)/(4) = -5, so the graph intersects the y-axis at y = -5.
Using this information, we can now sketch the graph of the function f(x).