Find all asymptotes, \( x \)-intercepts, and \( y \)-intercepts (if they do not exist, answer "DNE") for the graph of the rational function and sketch the graph of the function. \( f(x)=-\frac{2}{x-1} \) Asymptote(s): \( y= \)-intercept(s):

To analyze the function \( f(x) = -\frac{2}{x-1} \), we will find the asymptotes, \( x \)-intercepts, and \( y \)-intercepts. ### 1. Asymptotes **Vertical Asymptote:** Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). Set the denominator equal to zero: \[ x - 1 = 0 \implies x = 1 \] Thus, there is a vertical asymptote at: \[ x = 1 \] **Horizontal Asymptote:** To find the horizontal asymptote, we look at the behavior of \( f(x) \) as \( x \) approaches infinity. The degree of the numerator (which is 0) is less than the degree of the denominator (which is 1). Therefore, the horizontal asymptote is: \[ y = 0 \] ### 2. Intercepts **\( x \)-intercepts:** To find the \( x \)-intercepts, we set \( f(x) = 0 \): \[ -\frac{2}{x-1} = 0 \] This equation has no solution because a fraction is zero only when the numerator is zero, and the numerator here is \(-2\), which is never zero. Thus, there are no \( x \)-intercepts: \[ \text{\( x \)-intercepts: DNE} \] **\( y \)-intercept:** To find the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = -\frac{2}{0-1} = -\frac{2}{-1} = 2 \] Thus, the \( y \)-intercept is: \[ \text{\( y \)-intercept: } (0, 2) \] ### Summary of Results - **Vertical Asymptote:** \( x = 1 \) - **Horizontal Asymptote:** \( y = 0 \) - **\( x \)-intercepts:** DNE - **\( y \)-intercept:** \( (0, 2) \) ### Sketch of the Graph To sketch the graph of \( f(x) = -\frac{2}{x-1} \): 1. **Vertical Asymptote** at \( x = 1 \): The graph will approach this line but never touch it. 2. **Horizontal Asymptote** at \( y = 0 \): The graph will approach this line as \( x \) goes to \( \pm \infty \). 3. **\( y \)-intercept** at \( (0, 2) \): The graph will pass through this point. 4. The function is negative for all \( x \) except at the vertical asymptote, so the graph will be in the second quadrant (for \( x < 1 \)) and the fourth quadrant (for \( x > 1 \)). The graph will look like this: ``` | 2 | * | * | * | * | * ---|---------------- | * | * | * | * | * ``` This sketch represents the behavior of the function around the asymptotes and intercepts.