Which gear's motion depends on the motion of other gear? How many teeth does each gear have?
Words: For each turn that Gear B completes, Gear A will rotate only half a turn. Function: y= 12x Graph:
Practice makes perfect
We need to find how the number of turns of Gear A relates to the number of turns of Gear B. The exercise provides a picture of the situation and tells us that Gear A turns in response to the cranking of Gear B.
Independent and Dependent Variables
Since the motion of Gear A depends on the motion of Gear B, the number of turns of Gear A will be represented by the dependent variable y. Then, the number of turns of Gear B will be represented by the independent variable x.
Gear
Variable
Dependency
A
y
Dependent
B
x
Independent
Turns of Gear B
Notice that Gear B has 8 teeth while Gear A has 16 teeth.
Since Gear B has half the teeth of Gear A, for each turn that Gear B completes, Gear A will rotate only half turn. Therefore, we can calculate the number of turns of Gear A, y, as half the number of turns of Gear B, x. The equation for this relationship is as follows.
y = 1/2x
We can graph this relationship by first finding some points by using a table of values. We will use the function rule we found before and some arbitrary x values.
x
1/2x
y=1/2x
2
1/2* 2
1
4
1/2* 4
2
6
1/2* 6
3
8
1/2* 8
4
To make the graph, we can start by plotting these points. The line joining them will represent our function.
