f(x)=[5x^3-4x^2-8x+9]/[2x^2-1x-3]
Find the equation of the non-vertical asymptote.
y =
Does f(x) intersect its non-vertical asymptote? (yes or no)
What is the smallest value of x at which f(x) intersects its non-vertical asymptote? ( Enter No in the question blank if you answered no above.)
I got the answer to the first two parts, y=2.5x-.75 and yes. But i don't know how to figure out the last part of the question. Please help :)
your y = 2.5x - .75 is correct, (obtained by division)
so now we have to intersect
y = [5x^3-4x^2-8x+9]/[2x^2-1x-3] and y = 2.5x - .75
[5x^3-4x^2-8x+9] = [2x^2-1x-3][2.5x - .75]
[5x^3-4x^2-8x+9] = 5x^3 - 1.5x^2 - 2.5x^2 + .75x - 7.5x + 2.25
-1.25x = -6.75
x = 5.4
yes, f(x) intersects its non-vertical asymptote at x = 5.4
Wolfram seems to agree with me
http://www.wolframalpha.com/input/?i=%5B5x%5E3-4x%5E2-8x%2B9%5D%2F%5B2x%5E2-1x-3%5D+%3D+2.5x+-+.75
To find the smallest value of x at which f(x) intersects its non-vertical asymptote, you need to determine the x-value(s) where the function equals its non-vertical asymptote. In this case, the non-vertical asymptote is y = 2.5x - 0.75.
To determine the x-value(s) where f(x) intersects the asymptote, you need to equate f(x) to the equation of the asymptote and solve for x. In other words, set:
(5x^3 - 4x^2 - 8x + 9) / (2x^2 - x - 3) = 2.5x - 0.75
First, multiply both sides of the equation by (2x^2 - x - 3) to eliminate the denominator:
5x^3 - 4x^2 - 8x + 9 = (2.5x - 0.75)(2x^2 - x - 3)
Next, distribute and simplify:
5x^3 - 4x^2 - 8x + 9 = 5x^3 - 2.5x^2 - 7.5x - 1.5x^2 + 0.75x + 2.25
Combine like terms:
5x^3 - 4x^2 - 8x + 9 = 5x^3 - 3.5x^2 - 6.75x + 2.25
Now, bring all terms to one side of the equation:
0 = -0.5x^2 - 1.25x - 6.75
This is a quadratic equation. To find the smallest value of x, you need to find the x-coordinate of the vertex of this quadratic.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a is the coefficient of x^2 (-0.5) and b is the coefficient of x (-1.25).
x = -(-1.25) / (2 * -0.5)
x = 1.25 / -1
x = -1.25
So, the smallest value of x at which f(x) intersects its non-vertical asymptote is x = -1.25.