Answer:
The graph of the function f(x) = (-x^2 + x + 20) / (x + 4) can be obtained by analyzing its properties and plotting some key points.
First, we can determine the domain of the function by noting that the denominator of the fraction cannot be zero. Therefore, the function is defined for all real numbers except x = -4.
Next, we can look at the behavior of the function as x approaches positive or negative infinity. Since the degree of the numerator is 2 and the degree of the denominator is 1, the function approaches negative infinity as x goes to positive or negative infinity. This suggests that there is a maximum point on the graph of the function.
To find the maximum point, we can take the derivative of the function and set it equal to zero:
f'(x) = (-2x^2 - 4x + 21) / (x + 4)^2
Setting f'(x) = 0 and solving for x, we get:
-2x^2 - 4x + 21 = 0
Solving this quadratic equation, we get:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = -2, b = -4, and c = 21. Plugging in these values, we get:
x = (-(-4) ± sqrt((-4)^2 - 4(-2)(21))) / 2(-2) = 1 or -10.5
Therefore, the maximum point occurs at x = 1, which is also the vertex of the parabola.
To find the y-coordinate of the vertex, we can plug in x = 1 into the function:
f(1) = (-1^2 + 1 + 20) / (1 + 4) = 4
Therefore, the maximum point on the graph of the function is (1, 4).
We can also find the x-intercept and y-intercept of the function by setting x = 0 and y = 0, respectively:
x-intercept: 0 = (-0^2 + 0 + 20) / (0 + 4) = 5
y-intercept: f(0) = (-0^2 + 0 + 20) / (0 + 4) = 5
Therefore, the x-intercept and y-intercept are (5, 0) and (0, 5), respectively.
Using this information, we can sketch the graph of the function as follows:
- The function has a vertical asymptote at x = -4.
- The function passes through the point (0, 5) and approaches the line y = -x as x goes to positive or negative infinity.
- The function has a maximum point at (1, 4).
- The function passes through the point (5, 0).
- The function is symmetric about the vertical line x = 1/2, which is the axis of symmetry of the parabola.