1.6: 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 1: 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.
Suppose we wanted to visualize Joseph’s total cost of riding at the amusement park. Using the table generated in Lesson 1.5, 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.
Example 2: Graph the function that has the following table of values.
\begin{align*}&\text{Side of the Square} && 0 && 1 && 2 && 3 && 4\\ &\text{Area of the Square} && 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. Because the length of a square can be fractional values, but not negative, 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.
Writing a Function Rule Using a Graph
In many cases, you are given a graph and asked to determine its function. 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.
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 3: 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 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*}
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.
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 Chapter 9, you will learn how to approximate an equation to fit this data using a graphing calculator.
Determining Whether a Relation Is a Function
You saw that a function is a relation between the independent and the dependent variables. It is a rule that uses the values of the independent variable to give the values of the dependent variable. A function rule can be expressed in words, as an equation, as a table of values, and as a graph. All representations are useful and necessary in understanding the relation between the variables.
Definition: A relation is a set of ordered pairs.
Mathematically, a function is a special kind of relation.
Definition: A function is a relation between two variables such that the independent value has EXACTLY one dependent value.
This usually means that each \begin{align*}x-\end{align*}value has only one \begin{align*}y-\end{align*}value assigned to it. But, not all functions involve \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. Each person in the class cannot be more than one height at the same time. This relation is a function because for each person there is exactly one height that belongs to him or her.
Notice that in a function, a value in the range can belong to more than one element in the domain, so more than one person in the class can have the same height. The opposite is not possible, one person cannot have multiple heights.
Example 4: Determine if the relation is a function.
a) (1, 3), (–1, –2), (3, 5), (2, 5), (3, 4)
b) (–3, 20), (–5, 25), (–1, 5), (7, 12), (9, 2)
Solution:
a) To determine whether this relation is a function, we must follow the definition of a function. Each \begin{align*}x-\end{align*}coordinate can have ONLY one \begin{align*}y-\end{align*}coordinate. However, since the \begin{align*}x-\end{align*}coordinate of 3 has two \begin{align*}y-\end{align*}coordinates, 4 and 5, this relation is NOT a function.
b) Applying the definition of a function, each \begin{align*}x-\end{align*}coordinate has only one \begin{align*}y-\end{align*}coordinate. Therefore, this relation is a function.
Determining Whether a Graph Is a Function
One way to determine whether a relation is a function is to construct a flow chart linking each dependent value to its matching independent value. Suppose, however, all you are given is the graph of the relation. How can you determine whether it is a function?
You could organize the ordered pairs into a table or a flow chart, similar to the student and height situation. This could be a lengthy process, but it is one possible way. A second way is to use the Vertical Line Test. Applying this test gives a quick and effective visual to decide if the graph is a function.
Theorem: Part A) A relation is a function if there are no vertical lines that intersect the graphed relation in more than one point.
Part B) If a graphed relation does not intersect a vertical line in more than one point, then that relation is a function.
Is this graphed relation a function?
By drawing a vertical line (the red line) through the graph, we can see that the vertical line intersects the circle more than once. Therefore, this graph is NOT a function.
Here is a second example:
No matter where a vertical line is drawn through the graph, there will be only one intersection. Therefore, this graph is a function.
Example 4: Determine if the relation is a function.
Solution: Using the Vertical Line Test, we can conclude the relation is a function.
For more information:
Watch this YouTube video giving step-by-step instructions of the Vertical Line Test. CK-12 Basic Algebra: Vertical Line Test (3:11)
Practice Set
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 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*}
In 13 – 16, determine if the relation is a function.
- (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
- (1, 1), (1, –1), (4, 2), (4, –2), (9, 3), (9, –3)
- \begin{align*}&\text{Age} && 20 && 25 && 25 && 30 && 35\\ &\text{Number of jobs by that age} && 3 && 4 && 7 && 4 && 2\end{align*}
- \begin{align*}&&x && -4 && -3 && -2 && -1 && 0\\ &&y && 16 && 9 && 4 && 1 && 0\end{align*}
In 17 and 18, write a function rule for the graphed relation.
- 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
In 22 – 23, determine whether the graphed relation is a function.
Mixed Review
- A theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
- A group of students are in a room. After 25 students leave, it is found that \begin{align*}\frac{2}{3}\end{align*} of the original group are left in the room. How many students were in the room at the start?
- Evaluate the expression: \begin{align*}\frac{x^2+9}{y+2}, y = 3\end{align*} and \begin{align*}x=4\end{align*}.
- The amount of rubber needed to make a playground ball is found by the formula \begin{align*}A = 4 \pi r^2\end{align*}, where \begin{align*}r=radius\end{align*}. Determine the amount of material needed to make a ball with a 7-inch radius.
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