Learn About Linear Functions

TenMarks teaches you all about linear functions within math.

Learn About Linear Functions In this lesson, let’s learn about linear functions. And the problem given to us says that a water tanker is used to fill a community pool. The tanker pumps 600 gallons of water every hour. We need to write a linear function that describes the amount of water on the pool over time and we need to make a graph to show over the first 4 hours how much water is in the pool. And last we need to find the amount of water pumped in the pool after 6 hours. So let’s first understand what a linear function is. Linear function is a essentially a function whose result or when we plot the points in a coordinate plane is not a vertical line. It will not be a line that looks like this. A linear function is either represented as an equation or it is represented as a graph and we’ll see both. So let’s look at this particular problem. It says the tanker is used to fill the pool and the tanker pumps 600 gallons of water each hour. So in 1 hour it pumps 600 gallons of water. If the number of hours was x, how many gallons would it pumped? It would pump 600 times x. If the number of hours was x, in 1 hour it pumps 600 gallons so in x hours, it would be 600(x) gallons. That’s your linear function, where y is the output, y = 600x. This is x, let’s use the curve decks. Y = 600x, that’s our linear function or represent it as an equation. In order to graph it, here’s what I’m going to do. What I’m going to do is I’m going to create a table. So I’m going to put the value of x function is, the rule is 600(x). This is x, this is 600(x), this is the output y and we’re going to order the pairs. I’m going to make a quick table. It’s called the function table. When x = 1 hour, the rule is 600 and the value of x is 1. What’s the value of the output? How much water is being pumped? 600x1 which is 600. And the ordered pair is always 1 x, y. So it is 1, 600. Now when 2 hours passed, 600x would become 600 and 2, 600 multiplied by 2. Y would become 600(2) which is 1200. So the ordered pair would be 2, 1200. Similarly 3 and 4 would give 600, 3 and 600, 4. 600(3) is 1800, 600(4) is 2400 and the coordinate or the pair would be 3, 1800 and 4, 2400. Now that we’ve created the function table, let’s graph it. I have already drawn a graph. Notice what I did. We started 0, 0. This is x axis, number of hours. This is y axis which is amount of water. Look at the pairs; 1, 600. So I go 1 and up, I go to 600. That’s the first point. The second point is 2, 1200. That’s 2 on the x axis, 1200 on the y axis. Third point is 3, 1800, fourth point is 4, 2400; 4 on the x axis, 2400 on the y axis. Let’s go back the problem. We wrote the linear function that is right here. We made a graph, that’s done. And we need to find the amount of water pumped after 6 hours. I only drew the graph for 4 hours. So here’s the way to do it quickly. You just draw a straight line through these 4 points and then if we’re looking for after 6 hours, we just see what the measurement is. It is right here. What’s the answer? The answer is 3600. So after 6 hours, 3600 gallons have been pumped. All I did was look at 6, which number of hours on the x axis. See where the point was and I looked at the value across the y axis. Key thing to remember when we are given these functions again, a linear function is not a vertical line. As we can see in this case the line is a slope line it is not a vertical line. The way we create this function is we figure out how many gallons each hour which means the next hour, that 600(x) so we get the function. Once we have the function or the rule which is y = 600x, I can apply the rule to plug in values of x, apply the rule to get the values of y and create the ordered pairs. Once we have the ordered pairs, we can graph it. Once we have the few points, if it’s a straight line I can just draw a line through that and that would be the true value for any value of x and the corresponding values of y.