Join us for conversations that inspire, recognize, and encourage innovation and best practices in the education profession.
Available on Apple Podcasts, Spotify, Google Podcasts, and more.
Linear functions have come up in many situations so far in this session. In this section, we’ll consolidate some of our ideas about linear functions and look for connections between them. Note 14
Below are several input/output tables. For each table:
Note 15
Problem D1
Input | Output | |
1 | 0 | |
2 | 1 | |
3 | 2 | |
4 | 3 | |
5 | 4 |
Problem D2
Input | Output | |
1 | 1 | |
2 | 4 | |
3 | 9 | |
4 | 16 | |
5 | 25 | |
Tip: The recursive rule here includes n, the input number.
Problem D3
Input | Output | |
1 | 3 | |
2 | 5 | |
3 | 7 | |
4 | 9 | |
5 | 11 | |
Problem D4
Input | Output | |
1 | 9 | |
2 | 8 | |
3 | 7 | |
4 | 6 | |
5 | 5 |
Problem D5
Input | Output | |
1 | 5 | |
2 | 10 | |
3 | 15 | |
4 | 20 | |
5 | 25 | |
Problem D6
Input | Output | |
1 | 1 | |
2 | 1/2 | |
3 | 1/3 | |
4 | 1/4 | |
5 | 1/5 |
Problem D7
Input | Output | |
1 | -7 | |
2 | -7 | |
3 | -7 | |
4 | -7 | |
5 | -7 |
Problem D8
What are the characteristics of a linear function? How can you tell that a function is linear if you are given:
Note 14
The point of this section is to make connections between the different situations in which we’ve seen linear functions.
Note 15
Take 15-20 minutes to work on Problems D1-D7. Then think about the connections in the different representations, paying particular attention to instances where linear functions are different from other kinds of functions.
In Problem D6, the recursive rule for the function y = 1/x can be quite challenging. The easiest way to describe it is to use your input in the rule.
Some people consider a rule recursive only if it truly depends on previous outputs, with no reference to the input.
In summary:
Groups: Discuss the above statements.
Problem D1
The following solutions refer to the input variable as “x” and the output variable as “y.”
The closed-form rule is y = x – 1.
The recursive rule is y_{n} = y_{n-1} + 1, since the outputs grow by 1 each time.
This is a linear function, according to its graph, and the slope is 1.
Problem D2
The closed-form rule is y = x^{2}.
The recursive rule is harder to formulate for this one: it is y_{n} = y_{n-1} + (2n – 1). The key here is finding the pattern in the differences between each term. This is not a linear function.
Problem D3
The closed-form rule is y = 2x + 1.
The recursive rule is y_{n} = y_{n-1} + 2. Outputs grow by 2 each time.
This is a linear function, and the slope is 2.
Problem D4
The closed-form rule is y = -x + 10, or y = 10 – x (both are the same).
The recursive rule is y_{n} = y_{n-1} – 1. Outputs drop by 1 each time.
This is a linear function, and the slope is -1.
Problem D5
The closed-form rule is y = 5x.
The recursive rule is y_{n} = y_{n-1} + 5. Outputs grow by 5 each time.
This is a linear function, and the slope is 5.
Problem D6
The closed-form rule is y = 1 / x.
The recursive rule is very difficult. Two possible answers are 1 / y_{n} = 1 / y_{n-1} + 1, and y_{n}= y_{n-1} + 1 / (n)(n-1).
This is not a linear function. Notice that the rate of change is not constant.
Problem D7
The closed-form rule is y = -7.
The recursive rule is y_{n} = y_{n-1}, because every term is the same as the last.
This is a linear function, according to the graph, and the slope is 0 (which means it is a horizontal line).
Problem D8
If there is a closed-form rule for a function, and the function is linear, it will be in the form y = Mx + B, where M and B can be any real number — positive, negative, or 0. Note Problems D5 and D7, in which one of the two values is 0.
If there is a recursive rule given, it should be in the form y_{n} = y_{n-1} + M, where M is the slope of the line.
If a situation is described, it should involve a constant rate of change, such as a constant speed of a car, the constant slope of a ramp, or the constant price of gasoline per gallon.
If a table is given, the rate of change (change in output, divided by change in input) should always be the same number. If inputs are a sequence of numbers (like 1, 2, 3, 4, 5), the outputs should also form a sequence (3, 5, 7, 9, 11; 5, 10, 15, 20, 25).
If a graph is given, it should be a straight line (a linear function).