# 7.19: Linear and Non-Linear Function Distinction

**At Grade**Created by: CK-12

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

**. The values of a non-linear function do not form a straight line when they are graphed on a coordinate grid.**

*non-linear functions*
**How do we distinguish between a linear and a non-linear 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*} represents a function.

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

- Non-Linear 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.

### Image Attributions

Here you'll learn to distinguish between linear and non-linear functions.