Sketch the graph of the rational function y = rac(x+2)(x-2)(x+1) using vertical and horizontal asymptotes, intercepts, and interval testing.

Let's knock this out. To sketch the graph of the rational function y = ((x+2))/((x-2)(x+1)), follow these five steps: Step 1: Find the vertical asymptotes. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Set the denominator to zero: (x-2)(x+1) = 0 This gives x-2=0 or x+1=0. Therefore, the vertical asymptotes are at x=2 and x=-1. Step 2: Find the horizontal asymptotes. Compare the degree of the numerator (n) to the degree of the denominator (m). The numerator is x+2, so its degree is n=1. The denominator is (x-2)(x+1) = x^2 - x - 2, so its degree is m=2. Since n < m, the horizontal asymptote is y=0. Step 3: Find the x-intercepts. X-intercepts occur where y=0. This happens when the numerator is zero (and the denominator is non-zero). Set the numerator to zero: x+2 = 0 x = -2 The x-intercept is at (-2, 0). Step 4: Find the y-intercepts. Y-intercepts occur where x=0. Substitute x=0 into the function: y = ((0+2))/((0-2)(0+1)) y = (2)/((-2)(1)) y = (2)/(-2) y = -1 The y-intercept is at (0, -1). Step 5: Determine the behavior of the graph in intervals. Use the x-intercept (x=-2) and vertical asymptotes (x=-1, x=2) to divide the x-axis into intervals: (-, -2), (-2, -1), (-1, 2), (2, ). Test a point in each interval to determine the sign of y. For x < -2 (e.g., x=-3): y = (-3+2)/((-3-2)(-3+1)) = (-1)/((-5)(-2)) = (-1)/(10) < 0. The graph is below the x-axis. For -2 < x < -1 (e.g., x=-1.5): y = (-1.5+2)/((-1.5-2)(-1.5+1)) = (0.5)/((-3.5)(-0.5)) = (0.5)/(1.75) > 0. The graph is above the x-axis. For -1 < x < 2 (e.g., x=0): y = (0+2)/((0-2)(0+1)) = (2)/((-2)(1)) = -1 < 0. The graph is below the x-axis. For x > 2 (e.g., x=3): y = (3+2)/((3-2)(3+1)) = (5)/((1)(4)) = (5)/(4) > 0. The graph is above the x-axis. These points and behaviors allow you to sketch the curve, approaching the asymptotes and passing through the intercepts. Send me the next one 📸