# 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**.

- 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.

- 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.

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).

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).

The final answer should look like this…

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.

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

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

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.

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

Start from the center of the Cartesian plane.

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

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

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

**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.

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**.

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**.

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.

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.

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