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.
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.
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.
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.
The table gives us five sets of coordinate points: (0, 0), (1, 1), (2, 4), (3, 9), and (4, 16).
Graph the function that has the following table of values.
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).
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.
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. License: CC BY-NC 3.0
For 7-10, graph the function that has the following table of values.
x−10 −5 0 510y −3−0.524.57 Side of cube (in.)012 3Volume (in3) 01827 Time (hours) −2−10 1 2Distance from town center (miles)50 25 02550 x −2 −1 012y −400100200300800
To view the Review answers, open this PDF file and look for section 1.12.