Question: Determine all asymptotes of the following function:
$f(x) = \frac{-3x^2 - 14x + 49}{4x^2 - 9x + 2}$
(Note: The vertical asymptotes can be entered in any order.)
Round your answers to two decimal places.
The vertical asymptotes are $x = \underline{\hspace{2cm}}$ and $x = \underline{\hspace{2cm}}$.
The horizontal asymptote is $y = \underline{\hspace{2cm}}$.
1. **State the problem:** Find all vertical and horizontal asymptotes of the function $$f(x) = \frac{-3x^2 - 14x + 49}{4x^2 - 9x + 2}.$$
2. **Vertical asymptotes:** These occur where the denominator is zero (and numerator is not zero). Solve $$4x^2 - 9x + 2 = 0.$$
3. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=4$, $b=-9$, $c=2$.
4. Calculate the discriminant: $$\Delta = (-9)^2 - 4 \times 4 \times 2 = 81 - 32 = 49.$$
5. Find the roots:
$$x = \frac{9 \pm \sqrt{49}}{2 \times 4} = \frac{9 \pm 7}{8}.$$
6. Calculate each root:
$$x_1 = \frac{9 + 7}{8} = \frac{16}{8} = 2,$$
$$x_2 = \frac{9 - 7}{8} = \frac{2}{8} = 0.25.$$
7. **Vertical asymptotes:** $$x = 2$$ and $$x = 0.25.$$ Rounded to two decimals: $$x = 2.00$$ and $$x = 0.25.$$
8. **Horizontal asymptote:** For rational functions where degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients.
9. Leading coefficient numerator: $$-3$$, denominator: $$4$$. So,
$$y = \frac{-3}{4} = -0.75.$$
10. **Final answers:**
Vertical asymptotes: $$x = 2.00$$ and $$x = 0.25.$$
Horizontal asymptote: $$y = -0.75.$$