Plotting Points on a Graph

In this tutorial, I have prepared eight (8) worked-out examples on how to plot a point in a Cartesian plane (named in honor of French mathematician Renè Descartes). To plot a point, we need to have two things: a point and a coordinate plane.

Let’s briefly talk about each one.

A Point

A point in a plane contains two components where order matters! It comes in the form ([latex]x[/latex],[latex]y[/latex]) where [latex]x[/latex] comes first, and [latex]y[/latex] comes second.

The [latex]x[/latex]-value tells how the point moves either to the right or left along the [latex]x[/latex]-axis. This axis is the main horizontal line of the rectangular axis or Cartesian plane.

The [latex]y[/latex]-value tells how the point moves either up or down along the [latex]y[/latex]-axis. This axis is the main vertical line of the rectangular axis or Cartesian plane.

COORDINATE PLANE (Cartesian Plane)

A coordinate plane is composed of two lines intersecting at a 90-degree-angle (making them perpendicular lines) at the point (0,0) known as the origin.

here is an illustration of a coordinate plane where the main horizontal line is called the x-axis while the main vertical line is called the y-axis. the x and y-axes intersect at the point (0,0) which is commonly known as the origin. They are perpendicular to each other because they intersect at a 90-degree angle.

The [latex]x[/latex]-component of the point ([latex]x[/latex],[latex]y[/latex]) moves the point along a horizontal line. If the [latex]x[/latex]-value is positive, the point moves “[latex]x[/latex]-units” towards the right side. On the other hand, if the [latex]x[/latex]-value is negative, the point moves “[latex]x[/latex]-units” towards the left.

The [latex]y[/latex]-component of the point ([latex]x[/latex],[latex]y[/latex]) moves the point along a vertical line. If the [latex]y[/latex]-value is positive, the point moves “[latex]y[/latex]-units” in an upward direction. However, if the [latex]y[/latex]-value is negative, the point moves “[latex]y[/latex]-units” in a downward direction.

Quadrants of a Cartesian Plane

The intersection of the [latex]x[/latex]-axis and [latex]y[/latex]-axis results in the creation of four (4) sections or divisions of the Cartesian plane.

a coordinate plane can be divided into four quadrants. we start counting with the top half of the coordinate plane first. the top right section is considered the first (I) quadrant while the top left section is called the second (II) quadrant. we then continue by moving on to the bottom half of the coordinate plane. below the second quadrant is the third (III) quadrant located on the bottom left section of the plane. right next to it is the last or fourth (IV) quadrant which is located on the bottom right section of the Cartesian plane.

The first quadrant is located at the top right section of the plane.

The second quadrant is located at the top left section of the plane.

The third quadrant is located at the bottom left section of the plane.

The fourth quadrant is located at the bottom right section of the plane.

Examples of How to Plot Points on a Graph and Identify its Quadrant

Example 1: Plot the point (4,2) and identify which quadrant or axis it is located.

I will start by placing a dot at the origin which is the intersection of [latex]x[/latex] and [latex]y[/latex] axes. Think of the origin as the “home” where all points come from.

this is an illustration showing the location of the origin on the graph. when plotting points, we start by placing a dot at the origin which is where the x and y-axes intersect.

Next, I will move the dot from the origin 4 units to the right since [latex]x[/latex] = 4 (positive in [latex]x[/latex]-axis means right side movement). Remember, [latex]x[/latex]-value is the first number in the ordered pair (4,2).

in this illustration, we are shown where to move our pencil or hand from the origin to plot the point (4, 2). since our x-coordinate, which is 4, is positive, we move four units to the right from the origin.

From where I left off, I need to move two units going up, parallel to the main vertical axis since [latex]y[/latex] = 2 (positive in [latex]y[/latex]-axis means an upward movement). The [latex]y[/latex]-value is the second number in the ordered pair (4,2).

we are shown in this illustration that to plot the point (4,2) on the graph, we then move our hand or pencil upward from positive 4 which is our x-coordinate. this time, we move 2 units up since our y-coordinate which is 2, is positive. this is where you will place your dot.

The final answer should look like this…

the final location of the point (4, 2) is shown in this graph illustration. after placing the dot, you can write (4,2) to indicate that the point, from the origin, is located 4 units to the right of the origin along the x-axis and 2 units up from point (4,0) along the y-axis.

The point (4,2) is located in Quadrant I.

Example 2: Plot the point (–5, 4) and identify which quadrant or axis it is located.

Start by placing a dot at the origin which is known as the center of the Cartesian coordinate axis.

this image illustrates that on a graph, the location of the origin is where our x and y-axes intersect. this is where we should start when trying to plot a point.

From the origin, since [latex]x[/latex] = −5, move 5 units going left.

this illustration shows that to plot the point (-5,4), we move five units to the left of the origin, along the x-axis, because our x-coordinate is negative.

…followed by moving the point 4 units up because [latex]y[/latex] = 4.

after moving to the left of the origin, we then move 4 units up from the point (-5,0) since our y-coordinate is positive. we then mark our dot here.

This is the final answer. Since the plotted point is in the top left section of the [latex]xy[/latex]-axis, then it must be in Quadrant II.

this image illustrates where our point (-5,4) is located on the coordinate plane. as you can see, our dot is located 5 units to the left of the origin along the x-axis and 4 units up from point (-5,0) along the y-axis.

Example 3: Plot the point (5, –3) and identify which quadrant or axis it is located.

Start from the center of the Cartesian plane.

when plotting points, you start at the intersection of the x-axis and y-axis, which is also known as the origin.

Move 5 units to the right since [latex]x[/latex] = 5.

this graph shows that from the origin, we move five units to the right along the x-axis since our x-coordinate, which is 5, is positive.

Followed by moving 3 units down since [latex]y[/latex] = −3.

from the point (5,0), we then move 3 units down, since the y-coordinate of the point that we are trying to plot, which is (5,-3), is negative.

The final plotted point is shown below. Being in the bottom right section of the Cartesian plane means that it is in Quadrant IV.

the dot on this graph illustrates our point (5,-3) which is located 5 units to the right from the origin, along the x-axis, and 3 units down along the y-axis from the ordered pair (5,0).

Example 4: Plot the point (–2, –5) and identify which quadrant or axis it is located.

Place a dot at the origin (center of the [latex]xy[/latex]-axis). Since [latex]x[/latex] = −2, move the point 2 units to the left along the [latex]x[/latex]-axis. Finally, go down 5 units parallel to the [latex]y[/latex]-axis because [latex]y[/latex] = −5.

See the animated solution below.

this animated illustration shows our movement from the origin in order to plot the point (-2,-5). starting from the origin, we move 2 units to the left since our x-coordinate is negative. then we move 5 units down parallel to the y-axis from point (-2,0). this is because -5, which is our y-coordinate is negative as well. where we land is where our point (-2,-5) is located on the Cartesian plane. we can then write (-2,-5) to indicate or name the point.

The plotted point is located at the bottom left section of the Cartesian plane. Thus, it is in Quadrant III.

Example 5: Plot the point (0,3) and identify which quadrant or axis it is located.

I start by analyzing the given ordered pair. Since [latex]x[/latex] = 0, this means that there is no movement in the [latex]x[/latex]-axis. However, [latex]y[/latex] = 3 implies that I need to move it 3 units in the upward direction.

a point is moved 3 units up from the origin

The plotted point is neither in Quadrant I nor in Quadrant II. To describe its location, we say that it is found along the positive [latex]y[/latex]-axis.

Example 6: Plot the point (0, –4) and identify which quadrant or axis it is located.

This is very similar to example 5. There will be no movement along the [latex]x[/latex]-axis since [latex]x[/latex] = 0. On the other hand, [latex]y[/latex] = − 4 tells me that I need to move the point from the origin 4 units down.

a point is moved 4 units down from the origin (0,0)

The final point is located neither in Quadrant III nor Quadrant IV. I can claim that it is found along the negative [latex]y[/latex]-axis.

Example 7: Plot the point (–3,0) and identify which quadrant or axis it is located.

From the origin, I will move it 3 units to the left along the [latex]x[/latex]-axis since [latex]x[/latex] = −3. For [latex]y[/latex] = 0, it means no y-movement will follow.

a point is moved 3 units to the left from the origin (0,0)

The point is located neither in Quadrant II nor Quadrant III. It is found along the negative [latex]x[/latex]-axis.

Example 8: Plot the point (2,0) and identify which quadrant or axis it is located.

With [latex]x[/latex] = 2, I need to move it 2 units to the right. Having [latex]y[/latex] = 0 implies that no y-movement will occur.

a point is moved 2 units to the right from the origin

The plotted point is located neither in Quadrant I nor Quadrant IV. It is found along with the positive [latex]x[/latex]-axis.