7.19: Linear and NonLinear Function Distinction
Have you ever kept track of the number of books that you have read over a period of time? Look at Kendra.
Kendra and her friends have been reading books regularly.Everyone has a different reading rate, but Kendra is wondering if there is any connection between the number of books that she has read and the weeks of the contest. She has decided to include two of her friends.
Kendra kept track of her books and wrote down the following information. She included the information from two of her friends. Here is the number of books that the three girls read during each of the weeks recorded.
Week 1 = 3 books
Week 2 = 5 books
Week 3 = 4 books
Week 4 = 9 books
Kendra wonders if there is a connection between the values.
Is there?
This Concept is about linear and non  linear functions. You will know how to figure out the answer to this question by the end of the Concept.
Guidance
Up until this point, you have only been working with linear functions. Remember that a linear function will form a straight line when the values of the domain and the range are graphed on a coordinate grid. We can also have nonlinear functions. The values of a nonlinear function do not form a straight line when they are graphed on a coordinate grid.
How do we distinguish between a linear and a nonlinear function?
One of the easiest ways is to look at the graphs of the functions. Look at the two graphs below and you will see the difference between the two functions.
The first graph above shows a linear function because its graph is a straight line. The second graph shows a nonlinear function. Notice that the graph of this function is not a straight line. It is curved.
A nonlinear function does not have to look exactly like the function graphed above. Any function whose graph is not a straight line is a nonlinear function.
The equation \begin{align*}y=x^2\end{align*}
a. Graph that function on a coordinate plane.
b. Is the function linear or nonlinear?
First, consider part \begin{align*}a\end{align*}
Then use the equation to create a function table and find several ordered pairs for the function. Then you will be able to use the ordered pairs to graph the function.
You will need to use what you know about computing with integers and what you know about evaluating exponents to create the table.
\begin{align*}x\end{align*} 
\begin{align*}y\end{align*} 


2  4 
\begin{align*}\leftarrow y=x^2=(2)^2=(2) \cdot (2)=4\end{align*} 
1  1 
\begin{align*}\leftarrow y=x^2=(1)^2=(1) \cdot (1)=1\end{align*} 
0  0 
\begin{align*}\lnot \ y=x^2=(0)^2=0 \times 0=0\end{align*} 
1  1 
\begin{align*}\lnot \ y=x^2=(1)^2=1 \times 1=1\end{align*} 
2  4  \begin{align*}\lnot \ y=x^2=(2)^2=2 \times 2=4\end{align*} 
The ordered pairs shown in the table are (–2, 4), (–1, 1), (0, 0), (1, 1) and (2, 4).
Plot those five points on the coordinate plane. Then connect them. Notice that you cannot connect these points with a straight line. You will need to draw a curved line to connect them.
Consider part \begin{align*}b\end{align*} next.
Look at the function you graphed. The graph is curved. Since the graph is not a straight line, the equation \begin{align*}y=x^2\end{align*} represents a nonlinear function.
Now it's your turn. Look at each set of ordered pairs and determine whether or not each forms a linear or non linear function.
Example A
(0, 2)(1, 3)(2, 4)(3, 5)
Solution: Linear Function
Example B
(9, 6)(2, 7)(3, 5)(5, 9)
Solution: Non linear function
Example C
(10, 8)(8, 6)(6, 4)(4, 2)
Solution: Linear Function
Here is the original problem once again.
Kendra and her friends have been reading books regularly.Everyone has a different reading rate, but Kendra is wondering if there is any connection between the number of books that she has read and the weeks of the contest. She has decided to include two of her friends.
Kendra kept track of her books and wrote down the following information. She included the information from two of her friends. Here is the number of books that the three girls read during each of the weeks recorded.
Week 1 = 3 books
Week 2 = 5 books
Week 3 = 4 books
Week 4 = 9 books
Kendra wonders if there is a connection between the values.
Is there?
To figure this out, we can see if this is a linear or non  linear function. For there to be a linear, there will need to be a pattern between the week and the number of books.
Here is a graph of the function.
This is a non linear function. There isn't a connection between the week and the number of books read by the girls.
Vocabulary
Here are the vocabulary words in this Concept.
 Function
 A pattern where one element of from the domain is paired with exactly one element from the range.
 Function Rule
 the pattern rule for a function.
 Linear Function
 a function that forms a straight line when graphed
 NonLinear Function
 a function that does not form a straight line when graphed
Guided Practice
Here is one for you to try on your own.
Is this a linear or non linear function? Explain your answer. Then name the ordered pairs represented.
Answer
This is a linear function because the plotted points form a straight line when connected.
The ordered pairs graphed are (0, 2)(1, 4)(2, 6)(3, 8).
This is our answer.
Video Review
Here is a video for review.
 This is a James Sousa video on how to graph a linear function in the coordinate plane.
Practice
Directions: State if each graph shows a linear function or a nonlinear function.
1.
2.
3.
The table of ordered pairs below represents a function.
4. Plot those points on the coordinate plane below. Then connect those points to create the graph for this function.
5. Is the function you graphed a linear function or a nonlinear function?
The equation \begin{align*}y=\frac{x}{2}+4\end{align*} represents a function.
6. Complete the function table below to identify five ordered pairs for this function.
\begin{align*}x\end{align*}  \begin{align*}y\end{align*} 

4  
2  
0  
2  
4 
7. Plot those points on the coordinate plane below. Then connect those points to create the graph for this function.
8. Is the function you graphed a linear function or a nonlinear function?
The equation \begin{align*}y=x^2+2\end{align*} represents a function.
9. Complete the function table below to identify five ordered pairs for this function.
\begin{align*}x\end{align*}  \begin{align*}y\end{align*} 

2  
1  
0  
1  
2 
10. Plot those points on the coordinate plane below. Then connect those points to create the graph for this function.
11. Is the function you graphed a linear function or a nonlinear function?
The rule for a linear function is: add 1 to each \begin{align*}x\end{align*}value to find each \begin{align*}y\end{align*}value.
12. Write an equation to represent this linear function.
13. Graph the function on this coordinate plane.
The rule for a linear function is: multiply each \begin{align*}x\end{align*}value by 2 and then subtract 2 to find each \begin{align*}y\end{align*}value.
14. Write an equation to represent this linear function.
15. Graph the function on this coordinate 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 .Function Rule
A function rule describes how to convert an input value () into an output value () for a given function. An example of a function rule is .Linear Function
A linear function is a relation between two variables that produces a straight line when graphed.NonLinear Function
A nonlinear function is a function that does not form a line when graphed.Image Attributions
Description
Learning Objectives
Here you'll learn to distinguish between linear and nonlinear functions.