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Construct Functions to Model Linear Relationships

Sep 9, 2022
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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 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? 
  1. What is the slope formula? 
  1. 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 which can only have one output for each input. They are functions that can be represented by a straight-line graph. 
  1. The slope formula is, 

y = mx + b 

Where: 

“m” = the slope, 

parallel

“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.  

parallel

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. 

Write a function from a graph 

Solution: 

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

Solution:

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. 

 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?  

Interpret a function from a graph 

Solution: 

Solution: 

Exercise:

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

line containing points M and N
  1. Using the table of values, determine the equation of the line.
x y
0–9
1–6
2–3
30
43
  1. Look at the linear graph given below and write their function
linear graph
  1. Plot the given points on a graph and write their function.

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

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

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

  1. 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.

  1. The table shows the cost y (in dollars) of x pounds of groundnut seeds.
Pounds, xCost, y
22.80
3?
45.60

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

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

Draw a graph and write its function

  1. 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 

 Concept Map 

Comments:

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