What is the graph of the function f(x) = the quantity negative x squared plus x plus 20 over the quantity x plus 4

To graph the function f(x) = (-x^2 + x + 20) / (x + 4), follow these steps:

1. Analyze the numerator and factor it if possible. In this case, the quadratic expression -x^2 + x + 20 can be factored into -(x - 4)(x - 5).

2. Simplify the function by canceling out common factors in the numerator and denominator. There are no common factors between the factored numerator -(x - 4)(x - 5) and the denominator (x + 4), so the function stays the same.

3. Identify any vertical asymptotes by finding the values that make the denominator zero. Here, x + 4 = 0 leads to x = -4, which is a vertical asymptote since the factor x + 4 does not cancel with any factor in the numerator.

4. Identify any horizontal asymptotes by comparing the degrees of the polynomial in the numerator and the denominator. The degree of the numerator is higher (2) than the degree of the denominator (1), which means there is no horizontal asymptote; instead, the graph will have a slant asymptote which can be found using long division or polynomial division.

5. Identify the y-intercept by setting x = 0. So f(0) = (0^2 + 0 + 20) / (0 + 4) = 20 / 4 = 5. The y-intercept of the graph is (0,5).

6. Identify any x-intercepts by finding the roots of the numerator (setting the numerator equal to 0). This would be the values of x where the function equals zero. For the original function, the x-intercepts are where -x^2 + x + 20 = 0; in factored form, where (x - 4)(x - 5) = 0; so the x-intercepts are x = 4 and x = 5.

7. Plot the vertical asymptote, x-intercepts, y-intercept, and if possible, the slant asymptote. Then sketch the graph, knowing that the curve approaches the asymptotes and passes through the intercepts.

8. Since the leading coefficient of the quadratic is negative (-1), the parabola would open downwards. This means that as x goes towards infinity or negative infinity, f(x) would also go towards negative infinity.

The graph will resemble an upside-down parabola with a vertical asymptote at x = -4 and no horizontal asymptote.