 We can produce the graph of y equals f of x by finding ordered pairs x, y that satisfy the equation y equals f of x if we have the equation y equals f of x. But what if we don't? One possibility is we may be confronted with a real-world problem. And here's the thing to remember. Real-world problems require real-world knowledge. In the real universe, quantities change. A person's height changes as they get older. The wind speed changes during a hurricane. The temperature changes as you ascend a mountain. In a function, we consider how does the output change as the input changes? Does the output increase? Does the output decrease? Or does the output remain constant? One complicating factor is that the input can change in two ways. For example, your weight is a function of your age. Let's verify that if your age is the input and your weight is the output, at any given age t, your weight w of t has a unique value. And that's because you can't be two different weights at the same age. Remember the age t here is some real numbers, so we're talking about an exact instant in time at which you can't have two different weights. Now your change in weight can be described in two ways. Your weight was different, greater or lesser, when you were younger. Your weight will be different, greater or lesser, when you are older. And so to simplify things, we should always describe the change in the output as the input increases. So what about our graphs? To graph y equals f of x, we'll use our change descriptions. If our function is increasing as the output increases, remember equals means replaceable. Our function is f of x and f of x is y. Since equals means replaceable, anything we say about the function is something we can say about y. Meanwhile, the input is the x value. And so anything we want to say about the input is something we can say about x. And so the statement that our function is increasing as the input increases really means that the y values increase as the x values increase. Now increasing our x values moves us to the right, and so as we move right, we move up. If our function is decreasing as the input increases, then the y values decrease as the x values increase. So as we move right, we move down. If our function is constant as the input increases, the y values stay the same as the x values increase. So as we move right, we stay at the same level. So let's try to graph a function. Let h of t be the height of a person at age t years. Determine if h of t is a function, then describe how h of t changes. So if h of t is the height of a person at age t years, the person's age t is the input, and the person's height h of t is the output. So let's see if this is a function. At any given age, any input, a person can only have one height. So that means height is a function of age. And again, it's important to remember real-world problems require real-world knowledge. And that real-world knowledge will be important in order to determine what happens to the function. So remember, always describe the change in output as the input increases. And here, the person's age is the input, the person's height is the output. So we want to describe what happens as a person's age increases. Well, if you know something about the real universe, you know that as a person's age increases, their height will increase from birth, t equals zero, to when they stop growing, t equals sometime in their teenage years. Their height will be constant for most of their adult life, and at some point their height will start to decrease slightly. This is due to things like osteoporosis and things like that. And so we have an idea what happens to this height function. What if we want to sketch a graph of y equals h of t? We can use our information about what happens to h of t to sketch our graph. So if we want to summarize our changes, the height increases for a while, then stays the same for a while, then decreases slightly for a while. And remember, we are always describing these changes in terms of what happens as the input increases. And so this means as we move right, the height, the function, the y values, increase. We go up. If we continue to move right, the function, the y values, stay the same. And if we continue to go right, the function, the y values, decrease. And so our graph should look something like this. Now, it's important not to get too hung up on the details. This is supposed to be a sketch of the graph. So within limits, any graph that shows an increase for a while and then a constant section and then a slight decrease will also be a potential sketch of the graph. So our graph might look like this. Or we could have other growth patterns. And in the absence of additional details, we can't get much more precise. Or we might have more detailed information about what happens. Let a of t be the altitude of a plane t hours after it takes off. The plane takes one hour to reach its cruising altitude. It stays at this altitude for three hours, then increases its height to avoid turbulence. It stays at the increased height for one hour, then begins its descent to land one hour later. Let's sketch a graph of y equals a of t. And so we make a few observations. The plane's altitude increases for an hour as it climbs to reach its cruising altitude. The altitude is a function a of t, but in our graph a of t is the y value. And so our y values increase for the first hour. As we go right, we go up. The altitude remains the same for three hours. So as we go right, the y values stay the same. The height increases to avoid turbulence. It stays at this increased height for an hour, then descends to land one hour later.