graph all asymptotes of the rational function.\nf(x)=\frac{-x^{2}+x + 4}{x - 3}

Explanation:

Step1: Find the vertical asymptote.

Set the denominator equal to zero and solve for $x$. $$x - 3 = 0$$ $$x = 3$$ Check the numerator at $x=3$: $- (3)^2 + (3) + 4 = -9 + 3 + 4 = -2 \neq 0$. Thus, there is a vertical asymptote at $x = 3$.

Step2: Find the slant asymptote.

Compare the degree of the numerator (2) and the denominator (1). Since the degree of the numerator is exactly one greater than the degree of the denominator, there is a slant asymptote. Perform polynomial long division or synthetic division. Using synthetic division for $\frac{-x^2 + x + 4}{x - 3}$:

  3 | -1   1   4
    |     -3  -6
    ----------------
      -1  -2  -2

The quotient is $-x - 2$ and the remainder is $-2$. The function can be written as $f(x) = -x - 2 - \frac{2}{x - 3}$. As $x \to \pm \infty$, the term $\frac{2}{x - 3} \to 0$. The equation of the slant asymptote is $y = -x - 2$.

Answer:

The vertical asymptote is the line $x = 3$. The slant asymptote is the line $y = -x - 2$. To graph these, draw a vertical dashed line passing through $x=3$ on the x-axis, and draw a slanted dashed line with a y-intercept of -2 and a slope of -1.