Determine if the function below has a horizontal or a slant asymptote, and explain why. $$ f(x)=\frac{-12 x^{2}-10+23 x}{1-4 x} $$ Answer Attempt 1 out of 2 The function has a $$ \quad \vee $$ asymptote because the degree of the numerator is $$ \quad \vee $$ the degree of the denominator. As $$ x $$ approaches infinity (positive or negative), the value of $$ y $$ approaches $$ \begin{array}{l}\text {. }\end{array} $$

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**Complete Answer:**

The function has a **slant (oblique)** asymptote because the degree of the numerator is **one greater than** the degree of the denominator.

As $$ x $$ approaches infinity (positive or negative), the value of $$ y $$ approaches the line:
$$ y = 3x - 5 $$

**Explanation:**

1. **Determine the Degrees:**
   - **Numerator:** $$ -12x^{2} + 23x - 10 $$ has a degree of **2**.
   - **Denominator:** $$ 1 - 4x $$ has a degree of **1**.

2. **Identify the Asymptote Type:**
   - Since the degree of the numerator (**2**) is **one greater** than the degree of the denominator (**1**), the function has a **slant (oblique) asymptote**.

3. **Find the Slant Asymptote:**
   - **Perform Polynomial Long Division** of the numerator by the denominator:

$$
     \frac{-12x^{2} + 23x - 10}{-4x + 1}
     $$

- **First Term:** $$ \frac{-12x^{2}}{-4x} = 3x $$
     - **Multiply and Subtract:** 
       $$
       3x \times (-4x + 1) = -12x^{2} + 3x
       $$
       $$
       (-12x^{2} + 23x - 10) - (-12x^{2} + 3x) = 20x - 10
       $$
     - **Second Term:** $$ \frac{20x}{-4x} = -5 $$
     - **Multiply and Subtract:** 
       $$
       -5 \times (-4x + 1) = 20x - 5
       $$
       $$
       (20x - 10) - (20x - 5) = -5
       $$
     - **Result:** The quotient is $$ 3x - 5 $$ with a remainder of $$ -5 $$.

- **Slant Asymptote Equation:** $$ y = 3x - 5 $$

4. **Behavior as $$ x $$ Approaches Infinity:**
   - As $$ x $$ becomes very large (positively or negatively), the remainder term $$ \frac{-5}{-4x + 1} $$ approaches **0**.
   - Therefore, $$ f(x) $$ approaches the slant asymptote $$ y = 3x - 5 $$.

**Final Statement:**

$$
\boxed{
\begin{aligned}
\text{The function has a } & \text{slant asymptote because the degree of the} \
& \text{numerator is one greater than the degree of the denominator.} \
\text{As } x \text{ approaches infinity (positive or negative), the value of } y \text{ approaches } & y = 3x - 5.
\end{aligned}
}
$$
The function has a slant asymptote because the degree of the numerator is one greater than the degree of the denominator. As $$ x $$ approaches infinity, $$ y $$ approaches $$ y = 3x - 5 $$.

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