 So remember the graph of y equals f of x includes all points x, y that make the equation y equals f of x true. This means several things. First, any point x, y gives us a solution to f of x equals y. Points like h0, those are points on the x-axis, give us a solution to f of x equals 0. The point 0k, that's the y-intercept, gives us a solution to f of 0 equals y. And we can also use the graph to determine where f of x is increasing or decreasing. So remember that we describe the change in the function values as the input values increase. So if we're graphing y equals f of x, remember equals means replaceable. So our y values are the function values. So we want to describe how the y values change as we move to the right. If the y values are increasing, the function is increasing. If the y values are constant, the function is constant. And if the y values are decreasing, the function is decreasing. No big surprise there. Now, where there is a little bit of a surprise is when we describe the interval. Never include the beginning or ending points unless they are the beginning or ending of the graph. We'll explain that by way of an example. Well, how about this one? The graph shows the water level in a lake where t is the number of days since January 1, 2020. We want to find the period of time when the water level was increasing, decreasing, and constant. So there are two important things to remember. Equals means replaceable, and we always describe the change of a function as its input increases. So we want to find when the water level was increasing as the input values increased. Well, our water level is the function. Equals means replaceable, so we can replace water level with function. But in our graph, our function is the y values. So we can replace function with y values. Meanwhile, our input values are the t values. And what this means is we want to describe what happens as we move to the right. So we see that from t equals 0 to t equals 60, the y values, the water level, is increasing. And so we can say the water level is increasing in the interval from 0 to 60. Now remember, when describing an interval of change, you should exclude the end points unless they are the beginning or the end of the graph. And since 0 is the beginning of the graph, we do include 0 using the square brackets. However, since 60 is not the end of the graph, we exclude 60 and use a parenthesis. From t equals 60 to t equals 150, the water level was decreasing. And since neither 60 nor 150 is the beginning or the end of the graph, we want to exclude both end points. And so the interval is going to be between 60 and 150, excluding the end points, and so we'll use parenthesis. From t equals 150 to t equals 180, the water level was constant. And again, these are not end points, and so we're constant in the interval from 150, don't include it, up to 180, don't include it. From t equals 180 to t equals 210, the water level was increasing. So we can union that with our increasing interval. From t equals 210 to t equals 300, the water level was decreasing, so we'll union that with our decreasing interval. And from t equals 300 onward to the end of our graph, the water level was constant. And since 360 is the actual end of our graph, then we do include that in our interval, and we indicate that by using a square bracket at 360. Another thing we're often interested in is the extreme values of a function. These are the greatest or least values of the function. And to find these from the graph of y equals f of x, we look for the greatest or least value of a function. Again, equals means replaceable. Since y equals f of x are function, our function can be replaced with y. So those greatest or least values of the function are the greatest or least values of y. A further refinement is the difference between the local extreme value or the global extreme value. And one good way of thinking about this is if you're running a race, you win locally. You're faster than anyone else around you. Meanwhile, you set records globally. You're faster than anyone who has ever run the race. So if we go back to our water level graph, we see that the highest water level occurred at t equals 210. This is the greatest y value anywhere on the graph. But notice here at t equals 60, this y value is higher than anybody around it. It's a peak in the middle of the graph. And so we could call it a local maximum. How about the lowest level? So here at t equals zero, we have the lowest point anywhere on the graph. And a point like this, well, this is not a local minimum because there are points around it that are just as low. And so we have a maximum at t equals 210, a local maximum at t equals 60, and a minimum at t equals zero.