# Graphs and Functions

## Graphs of Functions


### Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.
An ordered pair  is a solution of a linear equation, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.
In , we can see that, in graph of the equation  for every x-value there is only one y-value, as shown in the accompanying table.
A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation  is a function.
If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.
If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.
This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

### Identify Graphs of Basic Functions

We used the equation  and its graph as we developed the vertical line test. We said that the relation defined by the equation  is a function.
We can write this as in function notation as  It still means the same thing. The graph of the function is the graph of all ordered pairs  where  So we can write the ordered pairs as  It looks different but the graph will be the same.
Compare the graph of  previously shown in  with the graph of  shown in . Nothing has changed but the notation.
As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.
Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if  is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.
We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is  In function notation, this linear function becomes  where m is the slope of the line and b is the y-intercept.
The domain is the set of all real numbers, and the range is also the set of all real numbers.
We will use the graphing techniques we used earlier, to graph the basic functions.
The next function whose graph we will look at is called the constant function and its equation is of the form  where b is any real number. If we replace the  with y, we get  We recognize this as the horizontal line whose y-intercept is b. The graph of the function  is also the horizontal line whose y-intercept is b.
Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.
The identity function,  is a special case of the linear function. If we write it in linear function form,  we see the slope is 1 and the y-intercept is 0.
The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.
The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.
The next function we will look at does not square or cube the input values, but rather takes the square root of those values.
Let’s graph the function  and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.
Our last basic function is the absolute value function,  Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.

### Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.
We will start by reading the domain and range of a function from its graph.
Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.
Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.
We are now going to read information from the graph that you may see in future math classes.

### Key Concepts

1. Vertical Line Test


2. Graph of a Function


3. Linear Function


4. Constant Function


5. Identity Function


6. Square Function


7. Cube Function


8. Square Root Function


9. Absolute Value Function



### Section Exercises


### Practice Makes Perfect

Use the Vertical Line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation.
In the following exercises, use the graph of the function to find the indicated values.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

### Chapter Review Exercises


### Graph Linear Equations in Two Variables

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, determine which ordered pairs are solutions to the given equations.
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
Find 
In the following exercises, find the x- and y-intercepts.
In the following exercises, find the intercepts of each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.

### Slope of a Line

Find the Slope of a Line
In the following exercises, find the slope of each line shown.
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
In the following exercises, graph the line of each equation using its slope and y-intercept.
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope-Intercept
Use Slopes to Identify Parallel and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

### Find the Equation of a Line

Find an Equation of the Line Given the Slope and 
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

### Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities in Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables

### Relations and Functions

Find the Domain and Range of a Relation
In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function:
ⓐ  ⓑ  ⓒ 
In the following exercises, evaluate the function.

### Graphs of Functions

Use the Vertical line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation
In the following exercises, use the graph of the function to find the indicated values.

### Practice Test

Find the slope of each line shown.
Graph the line for each of the following equations.
Find the equation of each line. Write the equation in slope-intercept form.
Graph each linear inequality.
In the following exercises, ⓐ graph each function ⓑ state its domain and range.Write the domain and range in interval notation.