Graficos com continuidade

(Video) Discrete vs. Continuous Relationships

by Friendly Math 101

Here's a video by Friendly Math 101 going over the differences between continuous and discrete graphs.

$\large{\text{Summary}}$

Discrete means something that cannot be broken down into fractions or decimals. An example would be people; you cannot have half a person. We typically say discrete things can be counted.

Continuous is something that can and must be able to be broken down into fractions and decimals. Continuous situations are said to be measured, we can measure them. An example of this would be height.

When figuring out if a graph is continuous or discrete we see if all the points are connected. If the line is connected between the start and the end, we say the graph is continuous. If the points are not connected it is discrete. Think about it, can you draw a line between the points? If you can, you've got a continuous function!

Let's practice by looking at the graphs she presents in the video!

Even though the function doesn't go out to infinity, the points are still connected and it's still a continuous function. Let's try another one!

Let's now look at some application examples of discrete and continuous things. As you can see continuous things can be measured. Time, temperature, weight, distance, and height are all measured.

Finally we're going to figure out if some applications are continuous or discrete relationships.

What is a continuous graph?

Continuous graphs have continuous lines. This makes them different from discrete graphs, which only include unconnected points.

Continuous graphs represent continuous data. These are data that cover whole intervals. Something like time can be broken down into infinitely small increments: hours, minutes, seconds, milliseconds, microseconds, nanoseconds, and on and on. This is why we can represent it as a line.

The same is true for things like weight. We can break that down into infinitely small increments: kilograms, grams, milligrams, micrograms, etc.

The height of all the people in an office, volume of gas left in a tank, and exact age of passengers on a plane are all examples of things that could provide continuous data. That's because each of the measurements can be broken down into extremely specific increments: we can measure height in fractions of an inch, volume in fractions of even milliliters, and age in days, hours, minutes, seconds....you get the idea.

On the other hand, you can never have half a squirrel living in a tree—the number of squirrels always has to be an integer, and there can never be anything in between. The same goes for deliveries, since a half-finished delivery might as well have just not been attempted.

Both A and B are continuous, since they don't have any holes or gaps in them! Technically, C doesn't have any holes or gaps in it either, but since it's only one point it doesn't have any space for there to be holes or continuity. So, only A and B are considered continuous. You could say that C is discrete, though.

Yep, both graphs can be considered continuous! Neither have any holes or gaps, and both look like smooth, completely connected lines.

Continuous graphs are really important when we evaluate functions in the future, including linear functions, quadratic functions, cubic functions, square root functions, and more!

What are Continuous Graphs?

Continuous graphs are graphs where there is a value of $y$ for every single value of $x$, and each point is immediately next to the point on either side of it so that the line of the graph is uninterrupted. In other words, if the line is continuous, the graph is continuous. For example, the red line and the blue line on the graph below are continuous. The green line is discontinuous.

The red and blue lines do not have any breaks in them. That means you can draw the red and blue lines without picking up your pencil. In contrast, you have to pick up your pencil to draw the green lines, so it is discontinuous.

The above lines are called functions, because there is one $y$ value for every $x$ value. The red line is a linear function, and the blue line is a non-linear function. To learn more about linear and non-linear functions, click here.

Let's look more closely at discontinuous graphs. The graphs below are all discontinuous (i.e., they are not smooth).

Graph A (blue) is discontinuous because there's simply a chunk missing. Graph B (red) is discontinuous because even though it APPEARS to be full, there's a point missing (denoted by the empty circle). Graph C (green) is also discontinuous, even though there is a value of $y$ for every value of $x$, since not all the values of $y$ right next to each other.

On the other side of things, if a graph doesn't have any lines on it at all and is just made up of a bunch of separate points (discrete data), it can be called discrete. Learn more about how discrete graphs differ from continuous graphs!

Discrete and continuous data are not good or bad - they represent different types of numbers and measurements! Different types of values will need different types of graphs. When evaluating functions, you will be able to tell whether your graph will be a continuous or discrete graph.

You will often use continuous graphs to estimate values and solutions for functions.