Find the equation of the non-vertical asymptote. y=??
5x^3+5x^2+8x-7/-9x^2+9x+9
What is the smallest value of x at which f(x) intersects its non-vertical asymptote?
To find the equation of the non-vertical asymptote, we need to perform polynomial division on the given rational function. The degree of the numerator is 3 and the degree of the denominator is 2, so the long division will result in a polynomial of degree 1, which will represent the non-vertical asymptote.
First, let's perform polynomial long division:
-5/9x + 13/9
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-9x^2 + 9x + 9 | 5x^3 + 5x^2 + 8x - 7
To determine 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 f(x) intersects the non-vertical asymptote. Since the non-vertical asymptote is a straight line, the intersection point will have the same y-coordinate as the asymptote.
The equation of the non-vertical asymptote is y = -5/9x + 13/9.
To find the point of intersection, we set the y-coordinate of the graph equal to the y-coordinate of the asymptote:
-5/9x + 13/9 = -5/9x + 13/9
On both sides, the -5/9x terms can be canceled out, leaving us with:
0 = 0
This equation is always true, which means that the graph of f(x) intersects its non-vertical asymptote at every point with the same x-coordinate.
Therefore, there is no specific smallest value of x at which f(x) intersects its non-vertical asymptote; it intersects the asymptote at all values of x.