(-9x^3-6x^2-x+3)/(2x^2+5x+2)
I have figured out the equation for the slant asymptote = (33/4)-(9x/2) but this next ? is throwing me off. I have tried graphing and everything and it doesnt seem to work.
What is the smallest value of x at which f(x) intersects its non-vertical asymptote? (Leave this question blank if you answered no above.)
well, just plug it in:
33/4 - 9x/2 = (-9x^3-6x^2-x+3)/(2x^2+5x+2)
http://www.wolframalpha.com/input/?i=solve+33%2F4+-+9x%2F2+%3D+%28-9x^3-6x^2-x%2B3%29%2F%282x^2%2B5x%2B2%29
To find the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to find the x-coordinate of the point where the graph of the function intersects the slant asymptote.
First, let's review the equation of the function and its slant asymptote. The function is f(x) = (-9x^3 - 6x^2 - x + 3) / (2x^2 + 5x + 2). The slant asymptote is given by y = (33/4) - (9x/2).
To find the point where the graph of the function intersects the slant asymptote, we need to equate the two equations and solve for x.
(-9x^3 - 6x^2 - x + 3) / (2x^2 + 5x + 2) = (33/4) - (9x/2)
To solve this equation, we can first multiply both sides by (2x^2 + 5x + 2) to eliminate the denominators:
-9x^3 - 6x^2 - x + 3 = (33/4)(2x^2 + 5x + 2) - (9x/2)(2x^2 + 5x + 2)
Next, simplify and rearrange the equation:
-9x^3 - 6x^2 - x + 3 = (33/2)x^2 + (165/4)x + (33/2) - (18/2)x^3 - (45/2)x^2 - (18/2)x
Combine like terms:
-9x^3 - 6x^2 - x + 3 = -18x^3 - 45x^2 - 18x + (33/2)x^2 + (165/4)x + (33/2)
Bring all the terms to one side of the equation:
-9x^3 - 6x^2 - x + 3 + 18x^3 + 45x^2 + 18x - (33/2)x^2 - (165/4)x - (33/2) = 0
Combine like terms again:
9x^3 + 45x^2 + 17x + 3 - (33/2)x^2 - (165/4)x - (33/2) = 0
Now, we have a cubic equation. To find the smallest value of x at which f(x) intersects the non-vertical asymptote, we can use numerical methods such as graphing the equation or using calculator functions like "solve" or "root".
Alternatively, we can use the Rational Root Theorem to find possible rational roots of the equation and then check each root to see if it satisfies the equation. This can help us narrow down the possible values of x.
Once we have determined the possible values of x, we can substitute each value back into the original equation f(x) = (-9x^3 - 6x^2 - x + 3) / (2x^2 + 5x + 2) to verify if it intersects the non-vertical asymptote.
Unfortunately, since the equation is quite complex, it is not feasible to provide an exact solution without employing numerical methods or a computer program.