In this lesson, our instructor Vincent Selhorst-Jones teaches about Graphs and how to do them correctly. Hell go over how to interpret graphs, and different inputs and outputs. Youll learn about the graph as a location of a solution and why you should pay attention to axes. Should you use arrows or not? Whats the vertical line test? Vincent will also teach about domain and range in graphs. Lastly, youll learn how graphing calculators can help you and have the opportunity to practice with examples.
A graph visually represents a function or equation in math. It gives us an intuitive picture of how the function "works".
There are two main ways to interpret what a graph means:
Input ⇒ Output: The graph tells us what happens to each input value. "If I plug in some number for x, where will it go?" The input values are on the horizontal, the outputs are on the vertical.
Location of Solutions: We can also interpret a graph as the location of all solutions to the equation. The graph of an equation is made up of all the points that make the equation true.
Between these two options, it's usually best to interpret it as the first one: how inputs are mapped to outputs. This gives us an intuitive way to see what happens as we change input values. However, the other way will occasionally come in handy, so
don't forget about it entirely.
Pay attention to the axes! The axes tell you where the graph is and what scale it has. Knowing this is important if you want to interpret what the graph means. [This is also called the graphing window.]
In this course, we will not put arrows on the ends of our graphs. Instead, we assume we're all aware the graph keeps "going" past the edge. We won't use arrows because we know that most graphs are just a tiny window on a much larger function. [Caution: Some
teachers might still want you draw arrows on the ends of your graphs. If that's the case, do what they say as long as you're in their class.]
The easiest way to plot graphs is to plot points one-by-one. Make a table of values, calculate various inputs and outputs, then plot them on the graph. Once you have enough points to see the shape, draw it in.
Almost always, the plotted points will connect with curves. As you see more and more functions, you'll start to learn the various shapes. Use this knowledge to help you draw graphs accurately.
Anytime you're not sure how to draw in a graph, just plot more points. As you plot more points, you have more information. As you have more information, the picture becomes easier to see. This is always an option, even for the most confusing graphs.
We can tell if a graph is the graph of a function with the Vertical Line Test. If a vertical line can be drawn that crosses the graph at more than one point, it is not a function. Why? Because this means a single input is mapped to two outputs,
so it can't be a function.
The domain of a function is all the inputs that a function can accept. Thus, every point on the x-axis that the graph is above or below is in the domain. However, if you can draw a vertical line on an x-value and it does not cross the graph, then
that x is not in the domain. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
The range of a function is all the possible outputs a function can create. Thus, every point on the y-axis that the graph is left or right of is in the range. However, if you can draw a horizontal line on a y-value and it does not cross the graph,
then that y is not in the range. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
If you haven't already noticed it, this is a great time to point out that this course has an appendix that's all about graphing calculators. Check out the appendix to learn more about graphing calculators, where you can find some free options, what they're
good for, and how to use them.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.