Questions: Write an equation for the function graphed below. The y intercept is at (0,-0.5) y=

Write an equation for the function graphed below. The \(y\) intercept is at \((0,-0.5)\). Find the vertical asymptotes. The vertical asymptotes are at \(x=-1\) and \(x=2\). Thus, the denominator must have factors \((x+1)\) and \((x-2)\). Find the horizontal asymptote. The horizontal asymptote is \(y=0\), which means the degree of the numerator is less than the degree of the denominator. Find the \(x\) intercepts. The \(x\) intercept is at \(x=1\), so the numerator must have a factor of \((x-1)\). Find the \(y\) intercept. The \(y\) intercept is at \((0,-0.5)\). Construct the equation. Let the equation be \[ y = \frac{a(x-1)}{(x+1)(x-2)} \] Substitute \(x=0\) and \(y=-0.5\): \[ -0.5 = \frac{a(0-1)}{(0+1)(0-2)} \] \[ -0.5 = \frac{-a}{-2} \] \[ -0.5 = \frac{a}{2} \] \[ a = -1 \] Therefore, the equation is \[ y = \frac{-(x-1)}{(x+1)(x-2)} = \frac{1-x}{(x+1)(x-2)} = \frac{1-x}{x^2-x-2} \]

\(\boxed{y = \frac{1-x}{x^2-x-2}}\)

\(\boxed{y = \frac{1-x}{x^2-x-2}}\)