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# Increasing, decreasing, positive or negative intervals

CCSS.Math:

## Video transcript

what I hope to do in this video is look at this graph y is equal to f of X and think about the intervals where this graph is positive or negative and then think about the intervals when this graph is increasing or decreasing so first let's just think about when is this function when is this function positive well positive means that the value of the function is greater than a zero means that the value of the function this means that the function is sitting above the x-axis so it's sitting above the x-axis in this place right over here that I am highlighting in yellow and it is also sitting above the x-axis over here and if we wanted to if we wanted to write those intervals mathematically let's see let's say that this point let's say that this this point right over here is x equals a let's say that this right over here is x equals B and this right over here is x equals C then it's positive it's positive as long as X is between a and B at x equals a or at x equals B the value of our function is zero but it's positive when X is between a and B a and B or if X is greater than C X is we could write it there C is less than X or we could write that X is greater than C these are the intervals when when our function is positive let me write this f of X f of X positive when X is in this interval or this interval or that interval so when is f of X negative let me do some other color color f of X is going to be negative well it's going to be negative if X is less than a so this is if X is less than a or if X is between B and C then we see that f of X is below the x-axis f of X is down here so this is where it's negative so here or or X is between B or C B X is between B and C and I'm not saying less than or equal to because if we at at B or C the value of the function f of B is 0 f of C is 0 that's where we are actually intersecting the x axis so that's that was reasonably straightforward now let's ask ourselves a different question what is the function increasing or decreasing so when is f of X f of X in increasing well increasing one way to think about it is for every time that X is increasing then Y should be increasing or another way to think about it you have you have a you have a positive rate of change of Y with respect to X we could even think about it as imagine if you had a tangent line at any of these points if you had a tangent line at any of these points the slope of that tangent line is going to be is going to be positive but the easiest way for me to think about it is as you increase X you're going to be increasing Y so where is the function increasing well I'm doing it in blue so it's increasing right until we get to this this point right over here right until we get to that point over there and then it starts decreasing until we get to this point right over here and then it starts increasing again it starts it starts increasing again so let me let make some more let me make some more labels here so let's say that this this is x equals d and that this right over your x let me do that green color so let's say this is x equals d now it's not a DB but you get the picture and let's say that this is X is equal to X is equal to let me do it a little bit X is equal to e X is equal to e so when is this function increasing well its increasing if X is less than D X is less than D and I'm not going to say less than or equal to because right at x equals D it looks like just for that moment the slope of the tangent line looks like it would be it would be constant we're going from increasing to decreasing so right at D where neither increasing or decreasing but then we're also increasing so if X is less than D or X is greater than e or X is greater than E and where is f of X decreasing so f of X let me just in a different color when is let me pick a move so f of X decreasing decreasing well it's going to be right over here it's going to be right between D and E between x equals d and x equals E but not exactly at those points because it both of those points you're neither increasing nor decreasing but you see right over here as x increases as you increase your x what's happening to your Y if you go from this point and you increase your X what happened to your Y your Y has decreased you increase your X your Y has decreased you increase your X Y has decreased to increase X Y has decreased all the way until this point over here so f of X is decreasing for X between D and E so hopefully that gives you a sense of things notice these aren't the same intervals that we are the the intervals where we're positive or negative don't perfectly coincide with when we are increasing or decreasing so it's very important to think about these separately even though they kind of sound the same