### Functions on a Cartesian Plane

We represent functions graphically by plotting points on a **coordinate plane** (also sometimes called the **Cartesian plane**). The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at a point called the **origin**. The origin has this name because it is the “starting” location; every other point on the grid is described in terms of how far it is from the origin.

The horizontal number line is called the **\begin{align*}x-\end{align*}axis** and the vertical line is called the **\begin{align*}y-\end{align*}axis**. We can represent each value of a function as a point on the plane by representing the \begin{align*}x-\end{align*}value as a distance along the \begin{align*}x-\end{align*}axis and the \begin{align*}y-\end{align*}value as a distance along the \begin{align*}y-\end{align*}axis. For example, if the \begin{align*}y-\end{align*}value of a function is 2 when the \begin{align*}x-\end{align*}value is 4, we can represent this pair of values with a point that is 4 units to the right of the origin (that is, 4 units along the \begin{align*}x-\end{align*}axis) and 2 units up (2 units in the \begin{align*}y-\end{align*}direction).

We write the location of this point as (4, 2).

#### Plotting Points on a Cartesian Plane

Plot the following coordinate points on the Cartesian plane.

a) (5, 3)

b) (-2, 6)

c) (3, -4)

d) (-5, -7)

Here are all the coordinate points on the same plot.

Notice that we move to the right for a positive \begin{align*}x-\end{align*}value and to the left for a negative one, just as we would on a single number line. Similarly, we move up for a positive \begin{align*}y-\end{align*}value and down for a negative one.

The \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes divide the coordinate plane into four **quadrants**. The quadrants are numbered counter-clockwise starting from the upper right, so the plotted point for (a) is in the **first** quadrant, (b) is in the **second** quadrant, (c) is in the **fourth** quadrant, and (d) is in the **third** quadrant.

**Graph a Function From a Table**

If we know a rule or have a table of values that describes a function, we can draw a graph of the function. A table of values gives us coordinate points that we can plot on the Cartesian plane.

#### Graphing a Function Given a Table of Values

1. Graph the function that has the following table of values.

\begin{align*}& x \quad -2 \quad -1 \quad 0 \quad \ \ 1 \quad \ 2\\ & y \qquad 6 \qquad \ 8 \quad 10 \quad 12 \quad 14\end{align*}

The table gives us five sets of coordinate points: (-2, 6), (-1, 8), (0, 10), (1, 12), (2, 14).

To graph the function, we plot all the coordinate points. Since we are not told the domain of the function or given a real-world context, we can just assume that the domain is the set of all real numbers. To show that the function holds for all values in the domain, we connect the points with a smooth line (which, we understand, continues infinitely in both directions).

2. Graph the function that has the following table of values.

\begin{align*}& \text{Side of square} \ \quad 0 \quad 1 \quad 2 \quad 3 \quad 4\\ & \text{Area of square} \quad 0 \quad 1 \quad 4 \quad 9 \quad 16\end{align*}

The table gives us five sets of coordinate points: (0, 0), (1, 1), (2, 4), (3, 9), and (4, 16).

To graph the function, we plot all the coordinate points. Since we are not told the domain of the function, we can assume that the domain is the set of all non-negative real numbers. To show that the function holds for all values in the domain, we connect the points with a smooth curve. The curve does not make sense for negative values of the independent variable, so it stops at \begin{align*}x = 0\end{align*}, but it continues infinitely in the positive direction.

### Example

#### Example 1

Graph the function that has the following table of values.

\begin{align*}& \text{Number of balloons} \quad 12 \quad 13 \quad 14 \quad 15 \quad 16\\ & \text{Cost} \qquad \qquad \qquad \quad \ \ 41 \quad 44 \quad 47 \quad 50 \quad 53\end{align*}

This function represents the total cost of the balloons delivered to your house. Each balloon is $3 and the store delivers if you buy a dozen balloons or more. The delivery charge is a $5 flat fee.

The table gives us five sets of coordinate points: (12, 41), (13, 44), (14, 47), (15, 50), and (16, 53).

To graph the function, we plot all the coordinate points. Since the \begin{align*}x-\end{align*}values represent the number of balloons for 12 balloons or more, the domain of this function is all integers greater than or equal to 12. In this problem, the points are not connected by a line or curve because it doesn’t make sense to have non-integer values of balloons.

In order to draw a graph of a function given the function rule, we must first make a table of values to give us a set of points to plot. Choosing good values for the table is a skill you will develop throughout this course. When you pick values, here are some of the things you should keep in mind.

- Pick only values from the domain of the function.
- If the domain is the set of real numbers or a subset of the real numbers, the graph will be a continuous curve.
- If the domain is the set of integers of a subset of the integers, the graph will be a set of points not connected by a curve.
- Picking integer values is best because it makes calculations easier, but sometimes we need to pick other values to capture all the details of the function.
- Often we start with one set of values. Then after drawing the graph, we realize that we need to pick different values and redraw the graph.

### Review (Answers)

For 1-5, plot the coordinate points on the Cartesian plane.

- (4, -4)
- (2, 7)
- (-3, -5)
- (6, 3)
- (-4, 3)
- Give the coordinates for each point in this Cartesian plane.

For 7-10, graph the function that has the following table of values.

- \begin{align*}& x \quad -10 \quad \ -5 \quad \ 0 \quad \ 5 \quad 10\\ & y \quad \ -3 \quad -0.5 \quad 2 \quad 4.5 \quad 7\end{align*}
- \begin{align*}& \text{Side of cube (in.)} \quad 0 \quad 1 \quad 2 \quad \ 3\\ & \text{Volume (in}^3) \qquad \ \ \ 0 \quad 1 \quad 8 \quad 27\end{align*}
- \begin{align*}& \text{Time (hours)} \qquad \qquad \qquad \qquad \quad \ -2 \quad -1 \quad 0 \quad \ 1 \quad \ 2\\ & \text{Distance from town center (miles)} \quad 50 \quad \ 25 \quad \ 0 \quad 25 \quad 50\end{align*}
- \begin{align*} x \quad \ -2 \quad \ \ -1 \quad \ \ \ 0 \quad \quad 1 \quad \quad 2\\ y \quad \ -400 \quad 100 \quad 200 \quad 300 \quad 800\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 1.12.