Question: Graph the function below. Use the given points to plot the horizontal asymptote, vertical…

Graph the function below. Use the given points to plot the horizontal asymptote, vertical asymptote, and x-intercept:

\[ y = -\frac{64}{(x + 5)} - 8 \]

Solution

Step 1: Identify the vertical asymptote. \[ y = -\frac{64}{x + 5} - 8 \] The denominator is zero when \( x + 5 = 0 \), so \( x = -5 \). Thus, the vertical asymptote is at \( x = -5 \). Step 2: Identify the horizontal asymptote. The horizontal asymptote is the constant term in the equation, which is \( y = -8 \). Step 3: Find the x-intercept. Set \( y = 0 \) and solve for \( x \): \[ 0 = -\frac{64}{x + 5} - 8 \] Add 8 to both sides: \[ 8 = -\frac{64}{x + 5} \] Multiply both sides by \( x + 5 \): \[ 8(x + 5) = -64 \] Expand and solve for \( x \): \[ 8x + 40 = -64 \] \[ 8x = -104 \] \[ x = -13 \] Thus, the x-intercept is at \( (-13, 0) \). Step 4: Summary of Key Features for Graphing - Vertical Asymptote: \( x = -5 \) - Horizontal Asymptote: \( y = -8 \) - x-intercept: \( (-13, 0) \) Use these points and asymptotes to plot the graph of the function.