Relations
Variables are letters such as , , , , that are used to keep track of quantities we want to measure or count. We are often interested in how two variables interact.
A relation describes how two variables are connected. We can represent a relation using words, a table, a graph, or an equation.
Independent variables: these are the values we choose; the values of the other variable in the relation depend on the value of this independent variable
Dependent variables: these are the values we calculate; they depend on the value of the independent variable
Examples
- An employee's daily pay VS the number of hours worked
- Independent variable: ________
- Dependent variable: ________
- The height of a tree VS the age of the tree
- Independent variable: ________
- Dependent variable: ________
- The time a student spends studying VS their final exam grade
- Independent variable: ________
- Dependent variable: ________
Linear Relations
A linear relation is where the dependent variable changes by an amount that is proportional to the change in the independent variable.
*This means that the rate of change (how fast the value of the variables are changing) is constant.
Example
A certain YouTuber gets paid $0.01 per view for her videos. Is the relation between the number of views and the total amount this YouTuber gets paid a linear relation?
Independent variable: ________
Dependent variable: ________
How do we tell if a relation is linear?
Table of Values The first differences in the table of values of a linear relation is constant
Graph The graph of a linear relation is a straight line
Equation The equation of a linear relation has a degree of 1
The independent variable is raised to an exponent of 1, so the equation has a degree of 1.
Answer
Therefore, this is a linear relation.
Example: Linear Relations
A company pays a base rate of $20 per day plus $10 per hour worked.
a) Identify the independent and dependent variables in this situation.
Independent variable: the number of hours worked.
Dependent variable: the amount of total pay.
b) Create a table of values for this situation.
c) Create the graph based on the table of values you create in part b).
d) Write the equation that represents this situation.
Let represent the total amount of pay in $.
Let represent the number of hours worked.
The equation is or .
e) Determine whether the relation between the number of hours worked and the amount of daily pay is linear or not.
This is a linear relation:
- Table of value: the first differences are constant
- The graph: the graph is a straight line
- The equation: the equation has a degree of 1