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:

\[ y = -\frac{32}{(x + 3)} + 8 \]

Solution

To analyze the function \[ y = -\frac{32}{x+3} + 8 \] we need to determine the horizontal asymptote, vertical asymptote, and x-intercept. Step 1: Find the vertical asymptote Vertical asymptotes occur where the denominator is zero. Set the denominator to zero and solve for \(x\): \[ x + 3 = 0 \\ x = -3 \] The vertical asymptote is \(x = -3\). Step 2: Find the horizontal asymptote For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator. Since the degree of the numerator (constant, 0) is less than the degree of the denominator (1), the horizontal asymptote is: \[ y = 0 \text{ (from calculation)} + 8 = 8 \] So the horizontal asymptote is \(y = 8\). Step 3: Find the x-intercept The x-intercept occurs where \(y = 0\). Set the function equal to zero and solve for \(x\): \[ 0 = -\frac{32}{x+3} + 8 \] Rearrange to find \(x\): \[ -\frac{32}{x+3} = -8 \] Multiply both sides by \(-(x+3)\): \[ 32 = 8(x+3) \] Divide both sides by 8: \[ 4 = x + 3 \] Solve for \(x\): \[ x = 1 \] The x-intercept is \(x = 1\). Summary: - Vertical asymptote: \(x = -3\) - Horizontal asymptote: \(y = 8\) - x-intercept: \(x = 1\)