Compare Linear and Non-Linear Functions

Key Concepts

• Compare two linear functions
• Compare a linear and a non-linear function
• Compare properties of linear functions

INTRODUCTION

In the earlier chapter, we learned about the representation of linear functions with an equation and a graph and the representation of non-linear functions with a graph. 

3.3.1. Compare two linear functions. 

Let us compare the two linear functions given below and find out which one has the greater rate of change. 

Step1: Find the constant rate of change for each linear function. 

Function A

Function B

Since 1.5 is greater than 1, the linear function B is greater than A. 

3.3.2 Compare a Linear and a Non-Linear Function 

Sharon started learning to ride a bicycle with her dad. When Sharon started bicycling, her dad noted her acceleration as follows: 

When Sharon’s younger sister started bicycling, her dad noted her acceleration of cycling and drew the following graph. 

Both the table and graph show the relationship between time and speed. Sharon’s dad wants to compare both his daughters’ acceleration in cycling. 

Sharon’s acceleration: 

We can see each input represents a unique value in the output. Hence, this relation is a function. Since it has a constant rate of change, it is a linear function. 

Sister’s acceleration: 

This relation is a function. Can you see 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. Hence, this is a non-linear function. 

3.3.3 Compare Properties of Linear Functions 

Let us compare the properties of the two linear functions given below and find their constant rate of change and initial values. 

Function A: 

Function B: 

y = 7x + 3 

Solution: 

Function A: 

Function B: 

Function A has a greater initial value, and function B has a greater rate of change. 

Exercise:

1. Identify whether the functions are linear or non-linear from the following tables:

Table A

Table B

2. Identify whether the functions are linear or non-linear from the following graphs:

3. Identify whether the functions are linear or non-linear from the following equations:
(a) y=x+4
b) y=0
c) y = x3 +1

4. Identity linear or non-linear:
(a)

(b) y= + 5

(c)

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

6. Plot the following points on a coordinate plane and find out linear or non-linear (2,7), (3,5),(5,9), and (9,6).

7. The equation y= x2 +2 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.

Is the function you graphed a linear function or a non-linear function?

8. The rule for a linear function is: to multiply each x-value by 2 and then subtract 2 to find each y-value. Write an equation that represents this linear function. Graph the function on this coordinate plane.

9. The graph shown below is that of the linear function that relates the value V (in $) of a car to its age, where t is the number of years after 2000.

Find the constant rate of change in the above graph.

10. Write the initial value and constant rate of change for the following linear functions:

(a) N(t) = 15t+200

(b)

(c)

What have we learned?

• Comparison of two linear functions.
• Comparing a linear equation with a non-linear function.
• Comparing initial values of two linear equations.
• Comparing the values of the constant rate of change of two linear equations.