SOLUTION: Graph {{{y=(3x)/(x^2-5x+6))}}}

Algebra ->  Graphs -> SOLUTION: Graph {{{y=(3x)/(x^2-5x+6))}}}      Log On


   

Question 138682: Graph y=%283x%29%2F%28x%5E2-5x%2B6%29%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y=%283x%29%2F%28x%5E2-5x%2B6%29%29 Start with the given function



Looking at the numerator 3x, we can see that the degree is 1 since the highest exponent of the numerator is 1. For the denominator x%5E2-5x%2B6, we can see that the degree is 2 since the highest exponent of the denominator is 2.


Horizontal Asymptote:

Since the degree of the numerator (which is 1) is less than the degree of the denominator (which is 2), the horizontal asymptote is always y=0

So the horizontal asymptote is y=0



--------------------------------------------------



Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

x%5E2-5x%2B6=0 Set the denominator equal to zero


Now let's use the quadratic formula to solve for x. If you need help with the quadratic formula, check out this solver.

After using the quadratic formula, we get the solutions
x=3 or x=2

So this means the vertical asymptotes are x=3 or x=2


--------------------------------------------------



X-Intercept(s)


y=%283x%29%2F%28x%5E2-5x%2B6%29%29 Start with the given function


0=%283x%29%2F%28x%5E2-5x%2B6%29%29 Plug in y=0

x=0 Solve for x


So the x-intercept is (0,0). Since there can only be one y-intercept, the y-intercept is also (0,0)

Now let's use this info to graph y=%283x%29%2F%28x%5E2-5x%2B6%29:


Graph of y=%283x%29%2F%28x%5E2-5x%2B6%29%29 with the horizontal asymptote y=0 (blue line) , the vertical asymptotes x=3 and x=2 (green lines), and the x and y intercept (0,0)