To find the horizontal and vertical asymptotes of the function $y(x) = \frac{6x^3 - 2}{2x^3 + 5x^2 + 7x}$, follow these steps:

1. Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero (since division by zero is undefined), but the numerator is non-zero at those points.

Set the denominator equal to zero and solve for $x$: $2x^3 + 5x^2 + 7x = 0$ Factor the equation: $x(2x^2 + 5x + 7) = 0$ This gives $x = 0$.

Now, solve $2x^2 + 5x + 7 = 0$ using the quadratic formula: $x = \frac{-5 \pm \sqrt{5^2 - 4(2)(7)}}{2(2)} = \frac{-5 \pm \sqrt{25 - 56}}{4} = \frac{-5 \pm \sqrt{-31}}{4}$ Since the discriminant is negative, there are no real solutions for $2x^2 + 5x + 7 = 0$. Therefore, the only vertical asymptote is: $x = 0$

2. Horizontal Asymptotes

Horizontal asymptotes depend on the degree of the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degrees are equal, the horizontal asymptote is $y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}}$.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique asymptote).

In this case, both the numerator and denominator are of degree 3. Therefore, the horizontal asymptote is: $y = \frac{6}{2} = 3$

Final Answer:

• Vertical asymptote: $x = 0$
• Horizontal asymptote: $y = 3$

Thus, the asymptotes are: $x = 0, y = 3$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the horizontal asymptote change if the degree of the numerator were greater than the denominator?
2. What happens to the behavior of the graph near vertical asymptotes?
3. Could the function have an oblique asymptote? If yes, how do you find it?
4. How do horizontal asymptotes affect the end behavior of a function?
5. What would happen if both the numerator and denominator had complex roots?

Tip: Always factor the denominator carefully to find the correct vertical asymptotes, and check the degree of terms to determine horizontal asymptotes.