Session 5 Solutions: Linear Functions

A B C D
Homework

Solutions for Session 5, Part D

See solutions for Problems: D1 | D2 | D3 | D4 | D5 | D6| D7 | D8

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.

<< back to Problem D1

Problem D2

The closed-form rule is y = x².
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.

<< back to Problem D2

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.

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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.

<< back to Problem D4

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.

<< back to Problem D5

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.

<< back to Problem D6

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).

<< back to Problem D7

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).

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Session 5: Index | Notes | Solutions | Video