What is the graph of the function f(x) = the quantity negative x squared plus x plus 20 over the quantity x plus 4? A) a line that crosses the x axis at negative 5 and the y axis at 5. Discontinuity exists at negative 4, 1. B) a line that crosses the x axis at 5 and the y axis at negative 5. Discontinuity exists at 4, negative 1. C) a line that crosses the x axis at negative 5 and the y axis at negative 5. Discontinuity exists at negative 4, 9. D) a line that crosses the x axis at negative 5 and the y axis at negative 5. Discontinuity exists at 4, negative 9.

The graph of the function f(x) = (−x^2 + x + 20) / (x + 4) is not a line; it is a rational function, which means it can have curves, asymptotes, etc. To understand its graph, let's first simplify it if possible.

We can factor the numerator as follows: −x^2 + x + 20 = −(x^2 - x - 20) = −(x - 5)(x + 4)

The function can then be rewritten as: f(x) = −(x - 5)(x + 4) / (x + 4)

We can see that (x + 4) is a common factor in both the numerator and denominator. If x ≠ −4, we can simplify the function to: f(x) = −(x - 5)

However, because we cannot divide by zero, there is a point of discontinuity where x = −4, which is not included in the domain of the original function. Thus, x = −4 is a vertical asymptote of the graph of f(x).

Now, f(x) simplifies to −(x - 5), which is a linear function with a slope of −1 and a y-intercept at 5 (because when x = 0, f(x) = −(0 - 5) = 5). The x-intercept occurs when f(x) = 0:

0 = −(x - 5) x = 5

So, the graph crosses the x-axis at x = 5 and the y-axis at y = 5.

The correct answer that describes the graph is none of the given options, but if we were to correct the options to match our findings, we would say: the graph is a line that crosses the x-axis at 5 and the y-axis at 5. Discontinuity exists at x = −4. There is no discontinuity at y = 1.

Extra: To further explain, let's cover some concepts:

1. The graph of a rational function, which is a function where one polynomial is divided by another, does not always look like a line. If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the graph will have a horizontal asymptote, usually at y=0. If the degrees are the same, the horizontal asymptote will be at the ratio of the leading coefficients.

2. Vertical asymptotes occur at points where the denominator is zero (and the numerator is not zero at those points), so the function is undefined. In this case, (x + 4) in the denominator causes a vertical asymptote at x = −4, which means the function goes to positive or negative infinity as x approaches −4.

3. Simplifying the rational function helps us see its true nature, like in this example where it simplifies to a line with a single point of discontinuity where the denominator was zero.

Remember, finding and factoring out common factors, determining x- and y-intercepts, and identifying asymptotes are important steps in understanding the behavior of a rational function's graph.