How is a sign chart used when trying to determine intervals where a function is positive and negative?
My teacher has asked us to determine the domain and zeros of a list of functions, and also use a sign chart to determine intervals where the function is positive and negative. I had no trouble identifying the domain and zeros; however, I am completely stumped when it comes to the sign chart. For example, the function f(x)=(x+5)(x-8). The domain is all real numbers, and the zeros are -5 and 8. But about the sign chart, I am beyond confused. Please help!
For the example `f(x)=(x+5)(x-8)`
(1) The domain is all real numbers
(2) The zeros are x=-5 and x=8
(3) The zeros divide the graph of the function into three intervals: from negative infinity to -5 (usually written as `(-oo,-5)` ), from -5 to x=8 ( or (-5,8) ) and from 8 to positive infinity ( or `(8,oo)` ).
You can plug in any value in the interval as a test value. So on the interval from negative infinity to -5, you can plug in any convenient value, say x=-6. Then `f(-6)=(-6+5)(-6-8))=(-1)(-14)=14>0` . Since `f(-6)>0` , all values in the interval are positive.
So for `x<-5,f(x)>0` or `f(x)>0` on `(-oo,-5)` .
Simarlily, you plug in a test value for `-5<x<8` , say x=0. Then `f(0)=(0+5)(0-8)=(5)(-8)=-40<0` .
So for `-5<x<8,f(x)<0` or `f(x)<0` on (-5,8)
Last, we plug in a test value for x>8, say x=9. Then `f(9)=(9+5)(9-8)=14(1)=14>0` so :
For x>8, f(x)>0 or f(x)>0 on `(8,oo)`
** The idea is that the function is continuous (naively you can draw the graph without lifting your pencil.). So the function stays positive (or negative) for all values between two consecutive zeros of the function.**