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.
![Write a function from a graph](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1293.png)
Solution:
Step1: Use a graph to represent the situation and to determine the slope.
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1294-1024x537.png)
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.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1295.png)
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.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1296.png)
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](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1297.png)
Solution:
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1305-1024x425.png)
Exercise:
1. Write the equation that represents the line containing points M and N in the following graph.
![line containing points M and N](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1306.png)
- 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
![linear graph](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1307.png)
- 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).
![light wave is a function](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1308.png)
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
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1309-1024x427.png)
![Construct Functions to Model Linear Relationships](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2022%2F09%2FConstruct-Functions-to-Model-Linear-Relationships.png&w=1920&q=50)
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