This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To sketch the graph of F(x) = (1)/(x^2 - 4), we follow these steps: Step 1: Find the vertical asymptotes and holes. Set the denominator equal to zero: x^2 - 4 = 0 (x-2)(x+2) = 0 This gives vertical asymptotes at x = 2 and x = -2. There are no common factors in the numerator and denominator, so there are no holes. Step 2: Find the horizontal or slant asymptotes. The degree of the numerator (0) is less than the degree of the denominator (2). Therefore, the horizontal asymptote is y = 0. Step 3: Find the x-intercepts. Set the numerator equal to zero: 1 = 0 This is impossible, so there are no x-intercepts. Step 4: Find the y-intercept. Substitute x = 0 into the function: F(0) = (1)/(0^2 - 4) = (1)/(-4) = -(1)/(4) The y-intercept is (0, -(1)/(4)). Step 5: Test for symmetry. Substitute -x into the function: F(-x) = (1)/((-x)^2 - 4) = (1)/(x^2 - 4) = F(x) Since F(-x) = F(x), the graph is symmetric about the y-axis. Step 6: Plot additional points. We have vertical asymptotes at x = -2 and x = 2, and a horizontal asymptote at y = 0. The y-intercept is (0, -1/4). Let's pick points in the intervals (-, -2), (-2, 2), and (2, ): • For x = -3: F(-3) = (1)/((-3)^2 - 4) = (1)/(9 - 4) = (1)/(5). Point: (-3, (1)/(5)). • For x = -1: F(-1) = (1)/((-1)^2 - 4) = (1)/(1 - 4) = -(1)/(3). Point: (-1, -(1)/(3)). • For x = 1: F(1) = (1)/((1)^2 - 4) = (1)/(1 - 4) = -(1)/(3). Point: (1, -(1)/(3)). • For x = 3: F(3) = (1)/((3)^2 - 4) = (1)/(9 - 4) = (1)/(5). Point: (3, (1)/(5)). Step 7: Sketch the graph. 1. Draw dashed vertical lines at x = -2 and x = 2. 2. Draw a dashed horizontal line at y = 0 (the x-axis). 3. Plot the y-intercept at (0, -(1)/(4)). 4. Plot the additional points: (-3, (1)/(5)), (-1, -(1)/(3)), (1, -(1)/(3)), (3, (1)/(5)). 5. Connect the points, ensuring the graph approaches the asymptotes. • For x < -2, the graph is above the x-axis and approaches y=0 as x - and x=-2 as x -2^-. • For -2 < x < 2, the graph is below the x-axis, passing through (0, -(1)/(4)), and approaches x=-2 as x -2^+ and x=2 as x 2^-. • For x > 2, the graph is above the x-axis and approaches y=0 as x and x=2 as x 2^+. The key features for sketching the graph are: • Vertical Asymptotes: x = -2, x = 2 • Horizontal Asymptote: y = 0 • X-intercepts: None • Y-intercept: (0, -(1)/(4)) • Symmetry: About the y-axis • Additional points: (-3, (1)/(5)), (-1, -(1)/(3)), (1, -(1)/(3)), (3, (1)/(5)) 3 done, 2 left today. You're making progress.