SOLUTION: Graph the rational function. f(x) = (1)/((x-4)(x-3))

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Question 201561: Graph the rational function.
f(x) = (1)/((x-4)(x-3))
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Unless you know some Calculus there is only a few things I can explain that will help you graph this function:
  1. The y-intercept is when x=0. f%280%29=1%2F%28%280-4%29%2A%280-3%29%29=1%2F12. So the y-intercept is (0, 1/12).
  2. Since the numerator has no x in it, there is no way for the numerator to become zero. And if the numerator of a fraction can't be zero, the fraction as a whole can't be zero. Since f(x) cannot be zero there are no x-intercepts. In other words the graph never crosses the x-axis.
  3. When x gets very large, positively or negatively, the denominator becomes a very large positive number. And a fraction with 1 in the numerator and a very large positive denominator is very small (close to zero) positive number (Think about it.) Therefore y=0 is the horizontal asymptote and the graph approaches y=0 from above at both ends.
  4. If x=4 or x=3 then the denominator would be zero. So x=4 and x=3 are vertical asymptotes for the function.
  5. The vertical asymptotes divide the graph into three sections:
    1. To the left of x = 3. If x is less than three, then (x-4) and (x-3) are both negative. Their product (and f(x)) would then be positive. So to left of x=3 the graph will be above the x-axis everywhere.
    2. Between x=3 and x=4. When x is between 3 and 4, (x-3) is positive and (x-4) is negative. So the product (and f(x)) will be negative. Therefore bewteen the vertical asymptotes the graph will be below the x-axis.
    3. To the right of x=4. Here both (x-4) and (x-3) are positive so both the product and f(x) will be positive. In other words, to the right of x=4 the graph will be entirely above the x-axis.
  6. Without Calculus the only other thing I can add is: Create a table of values and plot the points. Focus primarily on points near and between the vertical asymptotes. For example you might try the following x values: 2.9, 3.1, 3.5 3.9 and 4.1. But don't limit yourself to just these. Keeping in mind all of the above as you go, continue plotting points until you get an pretty good idea about how the graph must be.

I hope this helps. If you know some calculus let me know and I will explain wheat we can learn from the first and second derivatives that will help in determining the graph of this function.