 This lesson is on function attributes. We are given this unusual curve, and if we look at it, we come from negative infinity, turn up to a maximum here, come down with an asymptote, continue on from the asymptote towards another maximum, and then come around again another maximum, come down to a minimum, and there is a hole at two, then two lines, almost like an absolute value function, and then we go on to what looks something like a parabola and continue on towards positive infinity. Let's find out some of the attributes. What are the zeros of this function? Well, the roots or zeros of this function are when the function crosses the x-axis, and they are at negative 7.25, negative 6, negative 1, that's a hole, so we don't consider that one, 5, and again at 8.5. What are the points of discontinuity? Well, we have one point of discontinuity here, and that's at x is equal to 2, the point is 2-0. The other discontinuity is with the asymptote here, and that is a vertical asymptote of x is equal to negative 7. Now, let's look at where our function increases. It increases from negative infinity to negative 7.5, then from negative 7 all the way to negative 4, start again at negative 3 to negative 1.5, 0 to 2, and 4 to 7. And when we write these, we write them in parentheses each time. You'll see them written in parentheses, because when we use a maximum or a minimum, we say it is non-increasing and non-decreasing. What are the intervals over which our function decreases? Well, they are from negative 7.5 to negative 7, negative 4 to negative 3, negative 1.5 to 0, 2 to 4, 7 to infinity. What are the relative maxima? Our relative maxima are at negative 7.5, negative 4, negative 1.5, and 7. Or the points are negative 7.5 to negative 4, 3, negative 1.5, 2, and 7, 2 and 2 thirds. What are the relative or local minima? They are at x is equal to negative 3, 0, and 4. Or the points are negative 3 and 2 thirds, 0, negative 3, 4, negative 2. What is the absolute maximum on this? Well, our absolute maximum is this point here, and it's negative 4, 3. What is our absolute minimum? Well, we have none, because these arrows say that our function goes down to infinity there, and it also goes down to negative infinity there, so there is no absolute minimum. What is the equation for the horizontal asymptote? We do have a horizontal asymptote. It comes from going out towards negative infinity here, and as x goes out towards negative infinity, y is equal to 0. What is the equation of the vertical asymptote? And again, it is at x is equal to negative 7, right in here. What are the intervals over which our function is concave up? It is concave up from negative infinity to negative 7.5, again from negative 5, about here to negative 4, negative 4 to negative 2, negative 1 to 2. Where is our function concave down? It is concave down from negative 7.5 to negative 7, negative 7 to negative 5, negative 2 to negative 1, and 5 to infinity. So now where are our points of inflection? We have a point of inflection here at negative 7.5, because we are talking about changing from concave up to concave down. At negative 5, we have a point of inflection. It is a nice smooth curve in here, and it changes from concave down to concave up. At negative 2, we have a point of inflection because it changes from concave up to concave down. And at negative 1, we have a point of inflection because it changes from concave down to concave up. And again, we are going from a line and sort of a shop change here in our function so there's change in concavity on this last piece. This concludes our lesson on function attributes.