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Graph Quadrants: Examples & Definition

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  • 0:01 Graph Quadrants Defined
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Lesson Transcript
Instructor
Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

The x-axis and y-axis of a graph divide the graph into four quadrants. Learn about the four quadrants, how they are different from each other, and how they are useful in understanding mathematical patterns.

Graph Quadrants Defined

A 2-dimensional graph, Cartesian plane, includes negative and positive values of both x and y. This graph is divided into four quadrants, or sections, based on those values. The first quadrant is the upper right-hand corner of the graph, the section where both x and y are positive. The second quadrant, in the upper left-hand corner, includes negative values of x and positive values of y. The third quadrant, the lower left-hand corner, includes negative values of both x and y. Finally, the fourth quadrant, the lower right-hand corner, includes positive values of x and negative values of y.

In some ways, the quadrants across from each other diagonally are similar to each other. First, let's look at the first and third quadrants, using an example.

Understanding Quadrants One and Three

Imagine that you ask your family and friends how they feel about Facebook. They give you positive numbers if they like it and negative values if they dislike it. For example, your best friend might love Facebook. She gives it a 4. Your mother thinks Facebook is a complete waste of time. She gives it a -6. Your grandmother, on the other hand, doesn't even know what Facebook is. She is completely neutral and gives it a 0.

You do the same for how your family and friends feel about texting. They rate texting with a positive number if they like it and a negative number if they hate it. For example, your mother gives texting a 4 because she finds it a useful way to keep track of you.

Since you have two numbers for each person, you might graph how they feel on a Cartesian plane. How they feel about Facebook is x. How they feel about texting is y. For example, your mother's ratings are (-6,4). On a graph, it looks like this:

Graph with single point

When you start plotting everyone's responses, you find that some of them are consistent. They either love both Facebook and texting or they hate both.

Quadrant one is the 'love, love' section of the graph. In the graph, the pink and yellow faces are in quadrant one. These two individuals love both Facebook and texting. For both of these people, both ratings were positive numbers.

Quadrant three is the 'hate, hate' section of the graph. The blue and purple faces hate both Facebook and texting. For both of these people, both ratings were negative numbers.

Graph with points in quadrants 1 and 3

Quadrants one and three include all the points whose coordinates have the same sign - both positive or both negative.

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Additional Activities

How Graph Quadrants Relate to Inverse Functions

In the video lesson, we learned about the four quadrants of the graph. The first quadrant has positive x values and positive y values. The second quadrant has negative x values and positive y values. The third quadrant has negative x values and negative y values. The fourth quadrant has positive x values and negative y values. We learned in the lesson that the first and third quadrants are similar - both have coordinates of the same sign - and the second and fourth quadrants are similar - both have coordinates of opposite signs. We will learn what happens to points in the four quadrants when we look at the inverse of a function.

What is an Inverse Function?

Think of a function as a set of points (a,b). The inverse function can be thought of as the set of points (b,a). This means that if you have a point on the graph of your function, then to find a corresponding point on the graph of the inverse function you just flip-flop the coordinates. This is the same as reflecting the graph across the diagonal line y=x.

Example

  • Graph the points (4,5), (-2,4), (-3, -1), (7,-5). State what quadrants the points are in.
  • Graph the line y = x on your graph. Use a dotted line.
  • Graph the points of the inverse function corresponding to your original points. This can be done by reflecting across the line, y = x, or by simply interchanging the order of the coordinates of the points. State what quadrants these new points are in.

Solution

(4,5) is in quadrant 1 (labeled orange), (-2,4) is in quadrant 2 (labeled red) , (-3, -1) is in quadrant 3 (labeled blue) and (7,-5) is in quadrant 4 (labeled green).


Line y=x in black


(5,4) is in quadrant 1 (labeled orange), (4,-2) is in quadrant 4 (labeled red), (-1,-3) is in quadrant 3 (labeled blue) and (-5,7) is in quadrant 2 (labeled green).

Discussion

Why is it that points that were in quadrant 1 or 3 stayed in the same quadrant after reflecting? Why is it that points that were in quadrant 2 or 4 moved to quadrant 4 or 2 after reflecting?


Answer to Discussion Questions

Since points in quadrant 1 are of the form (positive, positive) then, after reflecting, they are still in the form (positive, positive), this means they remain in quadrant 1.

Since points in quadrant 3 are of the form (negative, negative) then, after reflecting, they are still in the form (negative, negative), this means they remain in quadrant 3.

Since points in quadrant 2 are of the form (negative, positive) then, after reflecting, they are in the form (positive, negative), this means they move to quadrant 4.

Since points in quadrant 4 are of the form (positive, negative) then, after reflecting, they are in the form (negative, positive), this means they move to quadrant 2.

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