Find the equation of the non-vertical asymptote. What is the smallest value of x at which f(x) intersects its?

-6x^3 + 2x^2 - 9x +3 / -5x^2 +9x - 4

1)Find the equation of the non-vertical asymptote.
2)What is the smallest value of x at which f(x) intersects its non-vertical asymptote?

Please help! Explain the process of each step thoroughly please and thank you!

for large x this looks like

-6 x^3/-5 x^2 = (6/5)x
which is a straight line of slope 6/5

I don't understand? Can you please explain?

To find the equation of the non-vertical asymptote, we first need to determine the degree of the numerator and the denominator of the rational function. In this case, the numerator is a cubic polynomial (-6x^3 + 2x^2 - 9x + 3) and the denominator is a quadratic polynomial (-5x^2 + 9x - 4).

1) Find the equation of the non-vertical asymptote:
a) If the degree of the numerator is less than the degree of the denominator, the non-vertical asymptote is y = 0. However, in this case, the degree of the numerator (3) is greater than the degree of the denominator (2). This means that there is no horizontal asymptote at y = 0.

b) To find the equation of the non-vertical asymptote, we can perform long division on the given rational function. Divide the denominator into the numerator to get the quotient.

-6x + 3
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-5x^2 + 9x - 4 | -6x^3 + 2x^2 - 9x + 3

-6x^3 + 9x^2 - 4x
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7x^2 -5x + 3

From the long division, we obtain the quotient -6x + 3. The non-vertical asymptote is given by this quotient.

Therefore, the equation of the non-vertical asymptote is y = -6x + 3.

2) To find the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to find the x-value(s) for which the rational function equals the equation of the asymptote.

Set the rational function equal to the equation of the non-vertical asymptote:

-6x^3 + 2x^2 - 9x + 3 / -5x^2 + 9x - 4 = -6x + 3

Multiplying both sides of the equation by (-5x^2 + 9x - 4) to remove the denominators, we get:

(-6x^3 + 2x^2 - 9x + 3) = (-6x + 3)(-5x^2 + 9x - 4)

Expanding the right side of the equation using distribution:

-6x^3 + 2x^2 - 9x + 3 = -30x^3 + 54x^2 - 24x + 15x^2 - 27x + 12

Combining like terms:

-6x^3 + 2x^2 - 9x + 3 = -30x^3 + (54x^2 + 15x^2) - (24x + 27x) + 12

Simplifying:

-6x^3 + 2x^2 - 9x + 3 = -30x^3 + 69x^2 - 51x + 12

Now, we need to solve this equation for x. To do this, we can subtract the right side from the left side, resulting in:

0 = -30x^3 + 69x^2 - 51x + 12 - (-6x^3 + 2x^2 - 9x + 3)

Simplifying further:

0 = -30x^3 + 69x^2 - 51x + 12 + 6x^3 - 2x^2 + 9x - 3

Combining like terms:

0 = -24x^3 + 67x^2 - 42x + 9

At this point, we have a polynomial equation that we can solve. However, finding the exact values of the solutions for this cubic equation can be quite complex, and we may need to use numerical methods or calculators to find an approximation.

Therefore, to determine the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to solve the equation -24x^3 + 67x^2 - 42x + 9 = 0. This can be done using numerical methods such as graphing the equation or using a calculator.