Final answer:
To plot the given rational function, firstly determine the vertical and horizontal asymptotes, then find the x-intercept and the y-intercept, and finally consider the behavior around these asymptotes. The graph will approach, but not touch, the vertical asymptote at x = 3.5 and the horizontal asymptote at y = -1/2.
Step-by-step explanation:
Plotting the Five Features of the Rational Function
To plot the five features of the given rational function f(x) = -(x + 3) / (2x - 7), we need to determine the following characteristics: Vertical asymptotes, horizontal asymptotes, x-intercepts, y-intercepts, and the behavior of the function near the asymptotes.
- Vertical Asymptote: Set the denominator equal to zero and solve for x: 2x - 7 = 0 which gives x = 7/2. So, there is a vertical asymptote at x = 3.5.
- Horizontal Asymptote: Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients: y = -1/2.
- X-Intercept: Set f(x) = 0 and solve for x: -x - 3 = 0, which results in x = -3.
- Y-Intercept: Find f(0): f(0) = -3/(-7) = 3/7.
- Behavior Near Asymptotes: Observe the sign of f(x) near the vertical asymptote and horizontal asymptote to understand how the graph approaches these lines.
The graph should show a curve that approaches the vertical line x = 3.5 without touching or crossing it, and extends towards the line y = -1/2 as x goes to positive or negative infinity. At x = -3, the graph will cross the x-axis, and at x = 0, the graph will cross the y-axis at y = 3/7.