f(x) = (9x^3 - 1x^2 + 8x + 4)/(-2x^2 - 4x - 2)
What is the smallest value of x at which f(x) intersects its non-vertical asymptote?
I took the slant asymptote which is -9/2x+19/2 and set it equal to the function and tried to solve it as a quadratic equation. But any of my answers I enter into my online homework are wrong apparently. Help?
Did you really mean to write -1x^2 as the second term of the numerator? or did you mean to write -19 x^2?
For very large x, the f(x) ratio becomes 9x^3/-2x^2 = -(9/2)x
That means the non-vertical asymptote is fasym(x) = -(9/2) x
At an intersection with the asymptote, you can subsitute (9/2)x for f(x).
If you make that substition and multiply both sides by (-2x^2 - 4x - 2), the cubic terms drop out and you are left wth a quadratic to solve. Is that what you did?
No, I wrote everything correctly.
Yes I did that and then I did multiply both sides by the denominator. The answer I got from the quadratic equation was not correct, however, according to my online homework.
To find the smallest value of x at which f(x) intersects its non-vertical asymptote, we need to find the x-coordinate at which the function crosses the slant asymptote or the oblique asymptote.
The slant asymptote of the function f(x) can be found by performing long division of the numerator by the denominator. Based on the given function, the slant asymptote is -9/2x + 19/2.
To find the x-coordinate at which f(x) intersects the slant asymptote, we need to set the function equal to the slant asymptote:
f(x) = -9/2x + 19/2.
Now, let's solve this equation to find the x-coordinate. Starting with the given function:
(9x^3 - x^2 + 8x + 4)/(-2x^2 - 4x - 2) = -9/2x + 19/2.
Multiply both sides of the equation by the common denominator (-2x^2 - 4x - 2) to clear the fractions:
(9x^3 - x^2 + 8x + 4) = (-9/2x + 19/2)(-2x^2 - 4x - 2).
Now, simplify and rearrange the equation to make it a quadratic equation:
9x^3 - x^2 + 8x + 4 = (-9/2)(-2x^3 - 4x^2 - 2x) + (19/2)(-2x^2 - 4x - 2).
9x^3 - x^2 + 8x + 4 = 9x^3 + 18x^2 + 9x - 19x^2 - 38x - 19 + 19x^2 + 38x + 19.
Cancel out like terms:
9x^3 - x^2 + 8x + 4 = 9x^3 - x^2 + 8x + 4.
Since both sides of the equation are identical, it means that this equation holds true for all values of x. In other words, f(x) intersects its slant asymptote at every x-value.
Therefore, there is no smallest value of x at which f(x) intersects its non-vertical asymptote.