Linear Functions Introduction

We model many relationships between quantities as a linear function or straight line.

Even if the relationship between two quantities is not a straight line, a straight line is a good model that allows good predictions in a small area. For example, the earth is round but using a flat model is useful for building things.

In a linear model for a given change anywhere in the independent variable (x-axis) there is the same change in the independent variable.
This is equivalent to saying the slope is the same anywhere in the relationship.

Linear relationships are defined by a straight line when graphed and allow easy prediction.

Implications of Linear Relationships

• What sorts of questions can you answer with a linear function?
• What is the significance of a change in the slope?
• Can part of a graph be linear and others not?

Diminishing Returns

For many relationships between quantities in nature, at some point the slope decreases.

We should be aware of where this point is if we want effective actions.

For example, say you have 24 hours until an exam. How many hours should you study? Imagine there is a positive slope at first with hours studying on the x-axis and the exam grade on the y-axis. At some point, more hours studying won’t raise the grade and could even lower it if you end up sleep-deprived.

Concepts

• Association/Correlation
• Independent variable
• Dependent variable
• Slope
• Proportional Relationship

Association/Correlation

• Two variables are correlated when the change in one variable results in a predictable change in the other variable.

Independent Variable

• The variable we can manipulate
• The variable we want to see the effect of changing
• Placed on the x-axis

Dependent Variable

• The variable we observe when we manipulate the independent variable
• Placed on the y-axis

Slope

This quantity relates a change in the independent variable to the change in the dependent variable.