the numerator will be equal to 0 when x = -5 and x = -1
the numerator is positive when x < -5
the numerator is negative when -5 < x < -1
the numerator is positive when x > -1
the denominator is x-4.
the denominator is negative when x < 4
the denominator is 0 when x = 4
the denominator is positive when x > 4
you have 4 critical points to look at.
they are -5, -1, 4.
the numerator is zero at x = -5 and -1.
the denominator is 0 at x = 4.
the intervals you want to look at are therefore:
x < -5
-5 < x < -1
-1 < x < 4
x > 4
since the functions are continuous, if they are above zero in an interval, they will stay above 0, and, if they are below zero in an intervalo, they will stay below 0.
choose test points in each interval.
i chose:
x = -6 for x < -5
x = -3 for -5 < x < -1
x = 1 for -1 < x < 4
x = 5 for x > 4
the results of my calculations are shown below:
- .5 for x = -6
+ .57 for x = -3
- 4 for x = 1
+ 60 for x = 5
this tells you that the function is ...
negative when x < -5
positive when -5 < x < -1
negative when -1 < x < 4
positive when x > 4
this can be seen graphically in the following two graphs.
the first graph shows an overall view.
the second graph zeroes in on the interval where the graph changes from negatie to positive and then to negative again.
the vertical asymptote is at x = 4
that's when the denominator of (x-4) is equal to 0.
a printout of my excel spreadsheet analysis is shown below:
the previous graphs showed the test points.
the following graphs show the intervals.
