**Representation of Functions**

### Key Concepts

- Representation of linear function with an equation and a graph
- Representation of non-linear function with a graph
- Identify functions from graphs

## Introduction

- In this chapter, we will learn to represent linear and non-linear functions.
- Identify whether a function is linear or non-linear based on the graph.

In the earlier chapter, we learned about the identification of functions.

### What is a Function?

A function is a relation that describes that there should be only one output for each input.

### Are all the relations functions?

The coffee shop menu shown in the picture consists of items and their prices.

Is price a function of the item?

Each item on the menu has only one price, so the price is a function of the item.

### Connect Representations of Functions:

Functions can be represented by equations, tables, symbols, or graphs.

Suppose a printing machine prints 100 lines at the start and gradually increases its printing speed by 15 lines per sec.

**Algebraic Representation:**

**Table Representation:**

**Graphical Representation:**

### Representation of Linear Function with an Equation and Graph

Ed works in a restaurant. He earns $8 per hour plus tips. One day he earned a total of $83, including $35 from tips. How many hours did he work that day?

Let us represent the linear function with an equation:

**Step1:** Interpret the given information to display as an equation

Ed worked for *x* hours on that day

He earned $83 including $35 from tips

His earnings are $8 per hour. So, the slope is 8

8*x* + 35 = 83

8*x* + 35 – 35 = 83 – 35

8X/8=48/8

*x *= 6

Let us represent the linear function with a graph:

If Ed wants to earn $147, how many more hours does he need to work?

**This is a linear function as the graph of the function is a straight line. **

By observing the graph, we can understand that Ed needs to work for 14 hours to earn $147.

Linear functions are functions that appear as straight lines when they are graphed.

## Representation of Non-Linear Function with a Graph

Ed needs to drive from Boston to Chicago, which is about 1,000 miles. The more time he has, the slower he can go. If he wants to get there in 20 hours, he needs to go 50 miles per hour because

1000/20 = 50. But if he can take 40 hours to get there, he only has to average 25 miles per hour, since

1000/40 = 25.

Here is a table that shows several times and speeds that satisfy the equation:

If we plot those points on a graph, it will be as follows,

This is a **non-linear function** as the graph of the function is not a straight line.

The rate of change in this graph is not constant. It shows a curved line.

Non-linear means the graph is not a straight line. The graph of a non-linear function is a curved line. A curved line is a line whose direction constantly changes.

### Identify Functions from Graphs

For many functions, we can obtain graphs. But not every graph will represent a function. The vertical line test can be used to determine whether a graph represents a function.

The following are the steps of the vertical line test to determine whether a graph represents a function:

**Step 1:** Draw a vertical line anywhere on the given graph.

**Step 2:** We have to check whether the vertical line drawn on the graph intersects the graph in at most one point.

**Step 3:** If the vertical line intersects the graph in at most one point, then the given graph represents a function.

If the vertical line intersects the graph at more than one point, then the given graph does not represent a function.

A graph represents a function only if every vertical line intersects the graph in at most one point; that is, each *x*-value corresponds to exactly one *y*-value.

## Exercise:

- The given equation represents the area of a square A as a function of its side length, l.

A = s^{2}

- Create a table of values that represents some inputs and their corresponding outputs and use the table of values to sketch a graph of the function.

- Does this function appear to be a linear function?

- It costs $6 to enter a carnival and an additional $4 for each activity you do inside the carnival. The equation below represents the total cost spent at the carnival,
*y*, as a function of the number of activities done,*x*.

*y* = 4*x* + 6

- Create a table of values that represents some inputs and their corresponding outputs and use the table of values to sketch a graph of the function.

- Does this function appear to be a linear function?
- What is the rate of change?
- What is the initial value?

- A chocolate factory makes 360 chocolates in 6 minutes. The total number of chocolates produced is a function of the number of minutes that the factory is running.
- Write an equation that describes the total number of chocolates produced (y) as a function of the number of minutes (x) that the factory runs.
- Use the equation you wrote in part (a) to find the number of chocolates produced when the factory runs for 3 minutes, 16 minutes, and 20 minutes.
- Plot the points from part b on the graph below.
- Is the function linear or non-linear? Explain how you know.

- The graph below shows a function. Do you think the function is linear or non-linear? Briefly justify your answer.

- Consider this graph of a line.

Write an equation for the line.

- In the given figure, there is a relationship between the number of blocks on one edge and the total number of blocks in each figure. Represent this relationship using a table, equation, and graph

- Use the vertical line test to determine whether the graph given below represents a function.

- Use the vertical line test to determine whether the graph given below represents a function.

- Use the vertical line test to determine whether the graph given below represents a function.

- As you inflate a balloon, its volume increases. The table below shows the increase in the volume of a round balloon as its radius changes.

Radius (in.) | Volume (in^{3}) |

1 | 4.19 |

2 | 33.52 |

3 | 113.13 |

4 | 268.16 |

5 | 523.75 |

Do you think a graph of the data would or would not be a straight line? You can make a graph to find out.

### What have we learnt:

- Representation of linear function with an equation.
- Representation of linear function with a graph.
- Representation of nonlinear function with a table.
- Representation of nonlinear function with a graph.
- Identify whether a function is linear or nonlinear based on a graph.

**Concept Map**

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: