Graph Quadrants | Properties & Examples

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.