For the function y=4x^2+5/x^2-6x+9, do both of the following:
a. State the vertical asymptote of the function and describe how you found it
b. State the horizontal asymptote of the function and describe how you found it
I will assume you meant:
y= (4x^2+5)/(x^2-6x+9)
y = (4x^2 + 5)/(x-3)^2
The denominator is zero when x = 3, so we have a vertical asymptote at x = 3
y= (4x^2+5)/(x^2-6x+9)
y= (4 +5/x^2)/(1 - 6/x + 9/x^2) , I divided top and bottom by x^2
Limit (4 +5/x^2)/(1 - 6/x + 9/x^2) as x ----> ± ∞
= 4
so we have a horizontal asymptote at y = 4
https://www.wolframalpha.com/input/?i=graph+y%3D+%284x%5E2%2B5%29%2F%28x%5E2-6x%2B9%29+from+-20+to+20
thank you for the help, although I wrote the problem exactly from what was on the assignment
When you write fractions on line, you must place brackets in the proper places to assure
the order of operation is maintained.
e.g. Just typing y=4x^2+5/x^2-6x+9 , would result in 5 ÷ x^2 as the only division
e.g. Here is how Wolfram would see it
https://www.wolframalpha.com/input/?i=y%3D4x%5E2%2B5%2Fx%5E2-6x%2B9
a. To determine the vertical asymptote of the function, we need to investigate the behavior of the function as x approaches certain values. The vertical asymptote occurs when the denominator of a rational function becomes zero.
In the given function, y = 4x^2 + 5 / (x^2 - 6x + 9). We need to find the values of x that make the denominator equal to zero.
The denominator, x^2 - 6x + 9, can be factored as (x - 3)(x - 3). So, (x - 3)^2 = 0.
Setting (x - 3)^2 = 0, we solve for x:
(x - 3)(x - 3) = 0
x - 3 = 0
x = 3
Therefore, when x = 3, the denominator becomes zero, which means there is a vertical asymptote at x = 3.
b. To find the horizontal asymptote of the function, there are a couple of cases to consider based on the degree of the numerator and denominator.
In this case, the degree of the numerator (4x^2 + 5) is greater than the degree of the denominator (x^2 - 6x + 9). Therefore, there is no horizontal asymptote.
When the degree of the numerator is greater, there is no horizontal asymptote, and the function will tend towards positive or negative infinity as x approaches infinity or negative infinity. In other words, the graph of the function will approach vertical lines as it extends outward.
So, for the given function y = 4x^2 + 5 / (x^2 - 6x + 9), there is a vertical asymptote at x = 3, and no horizontal asymptote.