1.13: Functions on a Cartesian Plane
Suppose that you have a set of points, where the \begin{align*}x\end{align*}-coordinates represent the number of months since you purchased a computer and the \begin{align*}y\end{align*}-coordinates represent how much the computer is worth. Would you know how to plot these points on a Cartesian plane? How about if the situation were reversed and you had the plotted points? Could you come up with the coordinates of the points and the function rule that would generate these points? In this Concept, you'll learn the skills necessary to perform tasks like these.
Guidance
Functions as Graphs
Once a table has been created for a function, the next step is to visualize the relationship by graphing the coordinates (independent value, dependent value). In previous courses, you have learned how to plot ordered pairs on a coordinate plane. The first coordinate represents the horizontal distance from the origin (the point where the axes intersect). The second coordinate represents the vertical distance from the origin.
To graph a coordinate point such as (4,2) we start at the origin.
Because the first coordinate is positive four, we move 4 units to the right.
From this location, since the second coordinate is positive two, we move 2 units up.
Example A
Plot the following coordinate points on the Cartesian plane.
(a) (5, 3)
(b) (–2, 6)
(c) (3, –4)
(d) (–5, –7)
Solution: We show all the coordinate points on the same plot.
Notice that:
For a positive \begin{align*}x\end{align*} value we move to the right.
For a negative \begin{align*}x\end{align*} value we move to the left.
For a positive \begin{align*}y\end{align*} value we move up.
For a negative \begin{align*}y\end{align*} value we move down.
When referring to a coordinate plane, also called a Cartesian plane, the four sections are called quadrants. The first quadrant is the upper right section, the second quadrant is the upper left, the third quadrant is the lower left and the fourth quadrant is the lower right.
Example B
Suppose we wanted to visualize Joseph’s total cost of riding at the amusement park. Using the table generated in a previous Concept, the graph can be constructed as (number of rides, total cost).
\begin{align*}r\end{align*} | \begin{align*}J(r) = 2r\end{align*} |
---|---|
0 | \begin{align*}2(0) = 0\end{align*} |
1 | \begin{align*}2(1) = 2\end{align*} |
2 | \begin{align*}2(2) = 4\end{align*} |
3 | \begin{align*}2(3) = 6\end{align*} |
4 | \begin{align*}2(4) = 8\end{align*} |
5 | \begin{align*}2(5) = 10\end{align*} |
6 | \begin{align*}2(6) = 12\end{align*} |
The green dots represent the combination of \begin{align*}(r, J(r))\end{align*}. The dots are not connected because the domain of this function is all whole numbers. By connecting the points we are indicating that all values between the ordered pairs are also solutions to this function. Can Joseph ride \begin{align*}2 \frac{1}{2}\end{align*} rides? Of course not! Therefore, we leave this situation as a scatter plot.
Writing a Function Rule Using a Graph
In this course, you will learn to recognize different kinds of functions. There will be specific methods that you can use for each type of function that will help you find the function rule. For now, we will look at some basic examples and find patterns that will help us figure out the relationship between the dependent and independent variables.
Example C
The graph below shows the distance that an inchworm covers over time. Find the function rule that shows how distance and time are related to each other.
Solution: Make a table of values of several coordinate points to identify a pattern.
\begin{align*}&\text{Time} && 0 && 1 && 2 && 3 && 4 && 5 && 6\\ &\text{Distance} && 0 && 1.5 && 3 && 4.5 && 6 && 7.5 && 9\end{align*}
We can see that for every minute the distance increases by 1.5 feet. We can write the function rule as:
\begin{align*}\text{Distance} = 1.5 \times \ \text{time}\end{align*}
The equation of the function is \begin{align*}f(x) = 1.5x\end{align*}.
In many cases, you are given a graph and asked to determine the relationship between the independent and dependent variables. From a graph, you can read pairs of coordinate points that are on the curve of the function. The coordinate points give values of dependent and independent variables. These variables are related to each other by a rule. It is important we make sure this rule works for all the points on the curve. Finding a function rule for real-world data allows you to make predictions about what may happen.
Analyze the Graph of a Real-World Situation
Graphs are used to represent data in all areas of life. You can find graphs in newspapers, political campaigns, science journals, and business presentations.
Example D
Here is an example of a graph you might see reported in the news. Most mainstream scientists believe that increased emissions of greenhouse gases, particularly carbon dioxide, are contributing to the warming of the planet. The graph below illustrates how carbon dioxide levels have increased as the world has industrialized.
From this graph, we can find the concentration of carbon dioxide found in the atmosphere in different years.
1900 - 285 parts per million
1930 - 300 parts per million
1950 - 310 parts per million
1990 - 350 parts per million
In future lessons, you will learn how to approximate an equation to fit this data using a graphing calculator.
Vocabulary
Quadrant: A quadrant is a section of the Cartesian plane. The quadrants are created by the intersection of the horizontal axis and vertical axis.
Scatter Plot: A scatter plot is a graph of coordinate points, showing the relationship between the independent and dependent variables.
Guided Practice
Graph the function that has the following table of values. Find the function rule.
\begin{align*}& 0 && 1 && 2 && 3 && 4\\ & 0 && 1 && 4 && 9 && 16\end{align*}
Solution:
The table gives us five sets of coordinate points:
(0, 0), (1, 1), (2, 4), (3, 9), (4, 16).
To graph the function, we plot all the coordinate points. We observe that the pattern is that the dependent values are the squares of the independent values. Because squaring numbers will always result in a positive output, and squaring a fraction results in a fraction, the domain of this function is all positive real numbers, or \begin{align*}x \ge 0\end{align*}. This means the ordered pairs can be connected with a smooth curve. This curve will continue forever in the positive direction, shown by an arrow.
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Functions as Graphs (9:34)
In 1 – 5, plot the coordinate points on the Cartesian plane.
- (4, –4)
- (2, 7)
- (–3, –5)
- (6, 3)
- (–4, 3)
Using the coordinate plane below, give the coordinates for a – e.
In 7 – 9, graph the relation on a coordinate plane. According to the situation, determine whether to connect the ordered pairs with a smooth curve or leave the graph as a scatter plot.
- \begin{align*}&& X && -10 && -5 && 0 && 5 && 10\\ && Y && -3 && -0.5 && 2 && 4.5 && 7\end{align*}
Side of cube (in inches) | Volume of cube (in inches\begin{align*}^3\end{align*}) |
---|---|
0 | 0 |
1 | 1 |
2 | 8 |
3 | 27 |
4 | 64 |
Time (in hours) | Distance (in miles) |
---|---|
–2 | –50 |
–1 | 25 |
0 | 0 |
1 | 5 |
2 | 50 |
In 10 – 12, graph the function.
- Brandon is a member of a movie club. He pays a $50 annual membership and $8 per movie.
- \begin{align*}f(x) = (x - 2)^2\end{align*}
- \begin{align*}f(x) = 3.2^x\end{align*}
- The students at a local high school took the Youth Risk Behavior Survey. The graph below shows the percentage of high school students who reported that they were current smokers. A person qualifies as a current smoker if he/she has smoked one or more cigarettes in the past 30 days. What percentage of high school students were current smokers in the following years?
(a) 1991
(b) 1996
(c) 2004
(d) 2005
- The graph below shows the average lifespan of people based on the year in which they were born. This information comes from the National Vital Statistics Report from the Center for Disease Control. What is the average lifespan of a person born in the following years?
(a) 1940
(b) 1955
(c) 1980
(d) 1995
- The graph below shows the median income of an individual based on his/her number of years of education. The top curve shows the median income for males and the bottom curve shows the median income for females (Source: US Census, 2003). What is the median income of a male who has the following years of education?
(a) 10 years of education
(b) 17 years of education
What is the median income of a female who has the same years of education?
(c) 10 years of education
(d) 17 years of education
quadrant
A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.Cartesian Plane
The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.Coordinate Plane
The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Image Attributions
Here you'll learn how to plot points generated from a function on a Cartesian plane. You'll also learn how to generate a table of values and a function rule by looking at the graph of a function.
Concept Nodes:
quadrant
A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.Cartesian Plane
The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.Coordinate Plane
The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .