2.2: Graphing on the Cartesian Coordinate Plane (2023)

The Cartesian Coordinate Plane

In the previous section, we talked about how we can use ordered pairs to form the building blocks of relations and functions. We introduced a few ways to understand these types of relationships between numbers, including lists of ordered pairs, mapping diagrams, tables, and in the case of functions, writing an equation. In this section, we look at a new way to understand the relationships between numbers by visualizing the ordered pairs on a graph.

To create a graph, we start with what is called the Cartesian Coordinate Plane. The Cartesian coordinate plane, shown below, uses a grid system to plot ordered pairs using two number lines at the same time called the \(x\)-axis and \(y\)-axis. The place these axes intersect is called the origin. For an ordered pair \((x, y)\), the \(x\)-axis displays the first coordinate by moving along the horizontal direction and the \(y\)-axis displays the second coordinate by moving along the vertical direction. For positive numbers, we move either right (for the \(x\) component) or up (for the \(y\) component). For negative numbers, we move either left (for the \(x\) component) or down (for the \(y\) component). The origin is located at the point (0, 0); this is where we always start before moving along the \(x\) or \(y\) axis to plot a point.

Let's look at some examples on how to plot points.

It is also useful to be able to look at a point on a graph and be able to identify the corresponding ordered pair.

Computing Distance between Two Points

It is useful to be able to talk about the distance between two points plotted on a graph. Suppose we have two points \((x_1, y_1)\) and \((x_2, y_2)\). We can use a formula based on the Pythagorean theorem to determine the distance between these two points.

Let's try this out with a few examples!

Computing the Midpoint of Two Points

We can also determine the midpoint for two points. That is, if we are given two points, it is possible to figure out a third point that is located exactly halfway between the two original points.

Using this formula, computing the midpoint of two points very straightforward. It's essentially finding their average \(x\) value and their average \(y\) value.

Graphing Relations and Functions

The biggest use for graphs is that it allows us to actually visualize the relations and functions that we are working with. In order to do this, we simply graph the ordered pairs in the relation or function on the Cartesian coordinate plane.

The process is pretty straightforward when we are given the ordered pairs to plot! But in the last section, we discussed how functions can be described by rules using equations, instead of a short list of ordered pairs. This means that in order to create ordered pairs to graph the function, we must choose our own inputs for this. For each input \(x\), the resulting ordered pair becomes

\((x, f(x)).\)

These ordered pairs will act as our ordered pairs \((x, y)\) that we use for plotting, where we treat \(f(x)\) as the \(y\) values.

We can choose a list of several convenient inputs that will help us determine the general shape of a function, collect inputs and outputs in a table, then create ordered pairs from them. These are what we will use to graph a function \(f(x)\) that we determine using an equation.

Key Concepts

• The Cartesian coordinate plane allows us to visualize ordered pairs by representing the inputs along horizontal number line called the \(x\) axis and outputs along a vertical number line called the \(y\) axis.
• How to plot an ordered pair
  1. Start at the origin (0, 0); this is where the axes intersect
  2. Move the number of units in the \(x\) component horizontally. If \(x\) is positive, move right. If \(x\) is negative, move left.
  3. Move the number of units in the \(y\) component vertically. If \(y\) is positive, move up. If \(y\) is negative, move down.
• We can compute the distance between two points \((x_1, y_1)\) and \((x_2, y_x)\) by using the equation

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\)

• We can compute the midpoint of two points \((x_1, y_1)\) and \((x_2, y_x)\) by using the equation

\(\left(\frac{x_2 + x_1}{2}, \frac{y_2 + y_1}{2}\right).\)

• To graph a function defined using an equation for its rule
  1. Create a table to keep track of inputs, outputs, and ordered pairs.
  2. Choose several inputs to use to create ordered pairs; convenient numbers such as -1, 0, 1 are good to include, and often times, 4-5 points is sufficient to get an idea of what the graph will look like. Add these inputs to the table.
  3. Compute the outputs \(f(x)\) corresponding to each input \(x\) by plugging the \(x\) value into the rule for \(f(x)\); add these to the table.
  4. Create ordered pairs from the inputs and their outputs; add to table
  5. Use the ordered pairs to plot the graph of the function.
  6. Draw the function by connecting the dots. Recall that when a function is defined by an equation, we have a lot of inputs for \(x\) to choose from. We account for this on the graph by sketching a picture of a graph suggested by the points plotted. Be sure to continue the graph beyond the last points to indicate that we could have used even more inputs!