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Linear and Non-Linear Function Distinction

Understand that linear functions form straight lines and non- linear functions don't form straight lines.

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Linear and Non-Linear Function Distinction

Let’s Think About It

The championship bicycling race is a major event. Jess has trained for years to compete at this level and can go quite fast. Starting from rest, Jess can accelerate to 20 miles per hour. The coach records times and speeds and puts them in a table (shown below). The coach wants to measure Jess’s progress and to see if there is a pattern in this data.

Input (time) Output (speed in mph)
0 0
1 2
3 5
5 10
8 20

Can you plot this data on a coordinate plane, and then determine if this data could be modeled by a linear function?

In this concept, you will learn to distinguish between linear and nonlinear functions.

Guidance

If you graph a linear function you will get a straight line. There are also nonlinear functions. If you graph the coordinates of a nonlinear function you will not get a straight line.

One of the easiest ways (but not the only way) to distinguish between a linear and a nonlinear function is to look at the graph of the function. Look at the two graphs below and you will see the difference between the two types of functions.

The first graph above is a linear function because its graph is a straight line. The second graph is a nonlinear function. Notice that the graph of this function is not a straight line. It is curved.

So, if you plot points from a function and cannot draw a straight line through them, then it is not a linear function.

Let’s look at an example.

The equation \begin{align*}y = x^2\end{align*}y=x2 is a function.

  1. Plot several points to sketch a graph of the function.
  2. Determine if the function linear or nonlinear.

First, consider part a.

Use the equation to create an input-output table. Use five \begin{align*}x\end{align*}x-values so you get a sense of what the function will look like when graphed. Pick positive and negative \begin{align*}x\end{align*}x-values centered around 0.

 \begin{align*}y=x^2\end{align*}y=x2 
Input \begin{align*}(x)\end{align*}(x)  Output \begin{align*}(y)\end{align*}(y)
 \begin{align*}-2\end{align*}2  \begin{align*}4\end{align*}4  \begin{align*}y = (-2)^2 = (-2) \cdot (-2) = 4\end{align*}y=(2)2=(2)(2)=4
 \begin{align*}-1\end{align*}1 \begin{align*}1\end{align*}1   \begin{align*}y = (-1)^2 = (-1) \cdot (-1) = 1\end{align*}y=(1)2=(1)(1)=1
 \begin{align*}0\end{align*}0  \begin{align*}0\end{align*}0  \begin{align*}y = (0)^2 = (0) \cdot (0) = 0\end{align*}y=(0)2=(0)(0)=0
 \begin{align*}1\end{align*}1  \begin{align*}1\end{align*}1  \begin{align*}y = (1)^2 = (1) \cdot (1) = 1\end{align*}y=(1)2=(1)(1)=1
 \begin{align*}2\end{align*}2  \begin{align*}4\end{align*}4  \begin{align*}y = (2)^2 = (2) \cdot (2) = 4\end{align*}y=(2)2=(2)(2)=4

The ordered pairs shown in the table are \begin{align*}(-2, 4), (-1, 1), (0, 0), (1, 1)\end{align*}(2,4),(1,1),(0,0),(1,1) and \begin{align*}(2, 4)\end{align*}(2,4).

Plot those five points on the coordinate plane. Then, draw a curve to connect them.

The graph of \begin{align*}y = x^2\end{align*}y=x2 is shown below. This kind of curve is called a parabola.

Next, consider part b.

Notice that you cannot connect these points with a straight line. You will need to draw a curved line to connect them.

The answer is \begin{align*}y = x^2\end{align*}y=x2 is not a linear function.

Guided Practice

Look at the graph of a function below. Is this a linear or nonlinear function? Explain your answer. Then, write the coordinates of the ordered pairs highlighted on the graph.

The answer is that this is a linear function because the graph is a straight line.

The ordered pairs graphed are \begin{align*}(0, 2), (1, 4), (2, 6), (3, 8)\end{align*}(0,2),(1,4),(2,6),(3,8).

Examples

Several points from a function have been provided in the examples below. Determine if the points could be part of a linear function.

Example 1

\begin{align*}(0, 2), (1, 3), (2, 4), (3, 5)\end{align*}(0,2),(1,3),(2,4),(3,5)

Plot these points on a coordinate plane. Since you can connect them with straight line (using a straight edge), they could be part of a linear function.

The answer is these points could be from a linear function.

Example 2

\begin{align*}(2, 7), (3, 5), (5, 9), (9, 6)\end{align*}(2,7),(3,5),(5,9),(9,6)

Plot these points on a coordinate plane. Because you cannot connect these lines with a straight line, these points are not part of a linear function.

The answer is these points are from a nonlinear function.

Example 3

\begin{align*}(4, 2), (6, 4), (8, 6), (10, 8)\end{align*}(4,2),(6,4),(8,6),(10,8)

Plot these points on a coordinate plane. Since you can connect them with straight line (using a straight edge), they could be part of a linear function.

The answer is these points could be from a linear function.

Follow Up

Remember the bike race?

Jess’s coach recorded times and speeds in the table below. You need to determine if this data could be modeled by a linear function.

Input (time) Output (speed in mph)
0 0
1 2
3 5
5 10
8 20

First, write the information in this table as coordinates: \begin{align*}(0,0), (1,2), (3,5), (5,10), (8,20)\end{align*}(0,0),(1,2),(3,5),(5,10),(8,20).

Then, plot these points (shown below).

License: CC BY-NC 3.0

Next, determine if a linear function could model this data. Can you draw a straight (not curved) line that connects all of these points? You can get close, but a straight line will not go through all of the points, and is likely not the best model for this data.

The answer is no, this data is not part of a linear function.

Video Review

https://www.youtube.com/watch?v=8KLDGlrjzaw&feature=youtu.be

Explore More

Determine if each graph shows a linear function or a nonlinear function.

1. 

License: CC BY-NC 3.0

2. 

License: CC BY-NC 3.0

3. 

License: CC BY-NC 3.0

4. The equation \begin{align*}y =\frac{x}{2}+4\end{align*}y=x2+4 is a function. Complete the table below to identify five ordered pairs for this function, and then plot the points on a coordinate plane. Then connect those points to sketch a graph for this function.

 \begin{align*}x\end{align*}x  \begin{align*}y\end{align*}y
-4
-2
0
2
4

License: CC BY-NC 3.0

5. Is the function you graphed in the previous question \begin{align*}\left( y =\frac{x}{2}+4 \right )\end{align*} a linear function or a nonlinear function?

6. The equation \begin{align*}y = x^2 + 2\end{align*} is a function. Complete the table below to identify five ordered pairs for this function. Plot those points on the coordinate plane below. Then connect those points to sketch a graph for this function.

 \begin{align*}x\end{align*}  \begin{align*}y\end{align*}
-2
-1
0
1
2

License: CC BY-NC 3.0

7. Is the function you graphed in the previous question \begin{align*}(y = x^2 + 2)\end{align*} 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.

8. Write an equation that represents this linear function.

9. Graph the function on this coordinate plane.

License: CC BY-NC 3.0

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.

10. Write an equation that represents this linear function.

11. Graph the function on this coordinate plane.

Vocabulary

Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of x, there is only one value for y.
Function Rule

Function Rule

A function rule describes how to convert an input value (x) into an output value (y) for a given function. An example of a function rule is f(x) = x^2 + 3.
Linear Function

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.
Non-Linear Function

Non-Linear Function

A non-linear function is a function that does not form a line when graphed.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0
  7. [7]^ License: CC BY-NC 3.0

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