Construct Functions to Model Linear Relationships

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. 

1. What is a linear equation? 

2. What is the slope formula? 

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

Answers: 

1. 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. 

2. 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 = 1.5x. 

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 = 4.25x + 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.

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

x	y
0	–9
1	–6
2	–3
3	0
4	3

3. Look at the linear graph given below and write their function

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

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

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

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

6. 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.

7. 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?

8. Write a linear function that represents the cost y of x pounds of seeds given in question number 7.
9. 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

10. 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