Question 1095062: find the asymptotes and intercepts of the function and graph the function.
g(x)= (x-2)/(x^2-2x-3)
Answer by greenestamps(13195)   (Show Source):
You can put this solution on YOUR website!
(1) Factor numerator and denominator as needed to get all linear factors:
(2) The x-intercept(s) are where g(x) is 0; g(x) is 0 when the numerator is 0. In your example, there is one x-intercept, at x=2. The only x-intercept is (2,0).
(3) The y-intercept is when x=0; evaluating g(0) gives y = 2/3. The y-intercept is (0,2/3).
(4) The vertical asymptote(s) are where the denominator is 0. In your example, that is at x=-1 and x=3.
(5) For large positive or negative values of x, denominator is much larger than the numerator, so the function value has a horizontal asymptote of y=0. Plugging in values shows that the function value is small positive for large positive values of x and small negative for "large negative" values of x.
In summary, we have this:
(1) function value is small negative for "large negative" values of x;
(2) vertical asymptote at x=-1;
(3) y-intercept at (0,2/3)'
(4) the only x-intercept at (2,0);
(5) vertical asymptote at x=3; and
(6) function value is small positive for large positive values of x
You can draw a good representation of the graph with pencil and paper using those constraints.
It should look like this graph, which is produced by software:
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