Graphing Basic Functions

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  • 0:06 Maps and Graphs
  • 2:24 Ordered Pairs and Quadrants
  • 4:14 Graphing Functions
  • 6:53 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Graphs are just like maps - when you know the language! Review how locations have x and y coordinates similar to latitude and longitude, and how to plot points in the Cartesian plane.

Maps and Graphs

Latitude and longitude help us to identify a single point on a map with just a set of numbers.
latitude and longitude

Have you ever compared maps from centuries ago to Google Earth? The guys back then were way off! I mean, look at this guy. He says that California is an island. Well, I guess they didn't really have GPS to give them the latitude and longitude when they were mapping coastlines. Latitude and longitude are important because they give the coordinates of every point on a map; they allow you to identify a single point with just a set of numbers: a latitude and a longitude. With a latitude and a longitude, you can identify any location on the globe. For example, San Francisco is 37 degrees north of the equator and 122 degrees west of England. There's no other city that's at exactly this latitude and longitude.

In math, we often use a Cartesian plane as our map, and x and y points instead of longitude and latitude. We draw some mathematical map with an x-axis - that's kind of like the equator - and a perpendicular y-axis - that's kind of like the Prime Meridian, that thing that goes through England. We can then plot any point on this graph, or map, by identifying its unique location, its longitude and latitude, if you will. So each point has an x-location, defined as the distance from the y-axis. The x-location is going to have a positive value if it is to the right of the y-axis and a negative value if it is to the left of the y-axis. The y-axis itself is at x=0. This point also has a y-location; this is the distance from the x-axis. This is like how far north you are from the equator. It's given a positive value if it's above the x-axis and a negative value if it's below the x-axis. This x-axis is defined to be at y=0. This means that we can plot any point by representing it in what's called an ordered pair: (x,y). For example, we can plot the point (5,4), which is an ordered pair. (5,4) is where x=5 and y=4. So I'm going to count 5 over from the y-axis, and I'm going to count 4 up from the x-axis.

A Cartesian plane is like a map containing x and y points instead of latitude and longitude.
cartesian plane

Ordered Pairs and Quadrants

You can do this with almost any point, like (3,1). That's 3 over to the right and 1 up. The point (-1,0) is 1 to the left and 0 up. Just remember that your first number is going to move you left and right, and your second number is going to move you up and down. The point (2,-2) is going to move me 2 to the right and 2 down. The point (0,0) is where the x and y axes meet. That has a really special name, the origin. Around the origin, separated by the x- and y-axes, are the four quadrants: I, II, III and IV.

The first quadrant is where both the x and y values are all positive; they're all greater than zero. So this would be like where the Northern Hemisphere meets the Eastern Hemisphere. In the second quadrant, all of the x values are negative, because we're on the left-hand side of the y-axis, and all the y values are positive, because we're above the x-axis. This is like where the Northern and Western Hemispheres meet. In the third quadrant, x < 0 and y < 0. This is like where the Southern and Western Hemispheres meet. Finally, where the Eastern and Southern Hemispheres meet, we have the fourth quadrant, where x > 0 and y < 0.

Usually we want to plot lines and curves, not just single points. Imagine the conflicts with Canada if we could only plot a few points to represent our border! In reality, there are an infinite number of points infinitely close to one another. So if I zoom in on the border, it might look like there are points, but those points touch. They're continuous, representing a line.

Graphing Functions

The 4 quadrants on a graph

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