Things You Need to Know
- A linear relation is a relation between two things in which when one thing changes in quantity by a certain fixed amount, the other thing changes by another certain fixed amount.
- In other words, suppose every time x goes up by 1, y goes up by 3. This is a linear relation.
- There are four ways to show linear relations (GOAT):
- Graphs
- Ordered Pairs
- Algebraically
- Tables
- The independent variable is the variable that is not directly influenced by a change in the other variable (the dependent variable, which is directly influenced by the independent variable).
- To determine which is which, ask yourself: "Does x depend on y, or does y depend on x?"
- Example: imagine comparing study time and test scores. Ask yourself: does your test score depend on how much time you spent studying, or does how much time you spend studying depend on what your test score was?
- From this, the first choice is true. So the test score depends on the study time, therefore the test score is the dependent variable and the study time is the independent variable.
- Example: imagine comparing study time and test scores. Ask yourself: does your test score depend on how much time you spent studying, or does how much time you spend studying depend on what your test score was?
- To determine which is which, ask yourself: "Does x depend on y, or does y depend on x?"
- For graphs, x values are placed along the horizontal axis and y values are placed along the vertical axis.
- x values are independent, y values are dependent.
- If you connect the points graphed with a line, we call this the "line of best fit".
- Ordered pairs are just the coordinates that you plot on the graph.
- x values are first, y values are second.
- Example: (4, -2)
- Algebraically refers to writing it as a linear equation. This is in the form:
- y = mx + b
- The "m" value is how much y changes by each time x goes up by 1. Also called the "slope".
- The "b" value is the value of y when x = 0. Also called the "y-intercept".
- Example: y = -3x + 5
- Every time x increases by 1, y decreases by 3.
- When x is 0, y is 5.
- y = mx + b
- Tables are just used to organize x values and y values.
- Linear relations can be referred to as having direct variation or partial variation.
- Direct variation means that there is no "b" value in the algebraic form. In other words, the graph passes through the origin (the point (0, 0)).
- Partial variation means there is a "b" value in the algebraic form.