**Construct Functions to Model Linear Relationships**

### Key Concepts

- Write a function from a graph
- Write a function from two values
- Interpret a function from a graph

## Introduction

- In this chapter, we will learn to compare two linear functions, compare linear function with a non-linear function.
- We will also learn to compare the properties of two linear functions.

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

- What is a linear equation?

- What is the slope formula?

- Find the constant rate of change and initial value of the linear equation y = 7x + 3

**Answers:**

- A linear function is a type of function that can only have one output for each input. They are functions that can be represented by a straight-line graph.

- The slope formula is,

y = mx + b

Where:

“m” = the slope,

“x” = input (x-value),

“b” = the y-intercept (where the graphed line crosses the vertical axis).

The slope (m) can be calculated as (change in y)/ (change in x)

m = (difference in y coordinates)/ (difference in x coordinates)

3.

**Construct Functions to Model Linear Relationships:**

### Write a function from a graph

**Example1:**

An ant walks at a constant speed and covers the distance of 1.5 feet in 1 second. Find the distance that it can cover if it walks at the same pace for 14 seconds.

**Solution:**

**Step1:** Use a graph to represent the situation and to determine the slope.

The slope of the line is the change in distance (*y*) divided by the change in time (*x*), which is

**Step2:** Use the slope to write an equation that represents the function shown in the graph.

The equation is y = 1.5*x.*

**Step3:** Use the equation to find the distance covered in 14 seconds.

*y* = 1.5(14)

*y* = 21

The ant can cover 21 feet in 14 seconds.

### Write a function from two values

**Example2:**

Jin is tracking how much food he feeds his dog each week. After 2 weeks, he has used 8 ½ cups of dog food. After 5 weeks, he has used 21 ¼ cups.

Construct a function in the form *y* = *mx* + *b* to represent the amount of dog food used, *y*, after *x* weeks.

**Solution:**

Step1: Find the constant rate of change.

The constant rate of change is 4.25.

**Step2:** Use the slope and one set of values for *x* and *y* to find the *y*-intercept.

21.25 = 4.25(5) + b

0 = b

The linear function that models this relationship is *y* = 4.25*x* + 0.

### Interpret a function from a graph

**Example3:**

The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing. What function represents the relationship?

**Solution:**

## Exercise:

1. Write the equation that represents the line containing points M and N in the following graph.

- Using the table of values, determine the equation of the line.

x | y |
---|---|

0 | –9 |

1 | –6 |

2 | –3 |

3 | 0 |

4 | 3 |

- Look at the linear graph given below and write their function

- Plot the given points on a graph and write their function.

(0, 2), (1, 3), (2, 4), (3, 5)

- Plot the given points on a graph and write their function.

(4, 2), (6, 4), (8, 6), (10, 8)

- Erik wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.

Write the ordered pairs, draw a graph and write its functions.

- The table shows the cost
*y*(in dollars) of*x*pounds of groundnut seeds.

Pounds, x | Cost, y |

2 | 2.80 |

3 | ? |

4 | 5.60 |

What is the missing y-value that makes the table represent a linear function?

- Write a linear function that represents the cost
*y*of*x*pounds of seeds given in question number 7. - The frequency
*y*(in terahertz) of a light wave is a function of its wavelength*x*(in nanometers).

Draw a graph and write its function

- A line passes through the points (1, 3) and (3, 7). Write a linear function in the form
*y = mx + b*for this line.

### What have we learned:

- Writing a function from a graph.
- Finding the constant rate of change and initial value from the given two values and writing its linear equation.
- Finding the constant rate of change and initial value from the given graph and writing its linear equation.

** 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: