1. **State the problem:** We need to analyze and graph the function $g(x) = -\frac{4}{x} + 3$. 2. **Identify key features:** This is a rational function with a vertical asymptote where the denominator is zero and a horizontal asymptote determined by the behavior as $x \to \pm \infty$. 3. **Vertical asymptote:** The denominator $x=0$ causes the function to be undefined, so there is a vertical asymptote at $x=0$. 4. **Horizontal asymptote:** As $x \to \pm \infty$, the term $-\frac{4}{x} \to 0$, so $g(x) \to 3$. Thus, the horizontal asymptote is $y=3$. 5. **Behavior near the vertical asymptote:** - For $x \to 0^+$, $-\frac{4}{x} \to -\infty$, so $g(x) \to 3 - \infty = -\infty$ (approaches $y=3$ from below in quadrant IV). - For $x \to 0^-$, $-\frac{4}{x} \to +\infty$, so $g(x) \to 3 + \infty = +\infty$ (approaches $y=3$ from above in quadrant II). 6. **Graph description:** The graph has two branches: - One in quadrant II, approaching $y=3$ from above and going to $+\infty$ near $x=0$. - One in quadrant IV, approaching $y=3$ from below and going to $-\infty$ near $x=0$. 7. **Axes labeling:** Label the $x$-axis and $y$-axis clearly without grid numbers. **Final answer:** The function $g(x) = -\frac{4}{x} + 3$ has vertical asymptote at $x=0$ and horizontal asymptote at $y=3$ with two hyperbola branches as described above.