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?

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.

Proportional Relationship

  • If b=0 we say x and y are proportional
  • Symbol \propto
  • Has a linear relationship
  • The independent variable is zero when the dependent variable is zero
  • If two things are proportional, the values of the two properties are related by a constant factor
  • Most of our unit conversions are proportional relationships

Drawing a Graph

When you draw a graph, think about what you are communicating to your audience.

  • What is the story you are trying to tell?
  • What is the range of data on the x and y axes?
  • Where should you place ticks and tick labels
  • Draw out axes, ticks, labels
  • Draw the data points on the graph
  • Decide if shapes, color, and size would help you
  • Draw the trendline
  • Draw any annotations

Linear equation

y = mx + b

Estimations

Estimations

Linear equation

Linear equations

  • If b \neq 0 it is a linear function.
  • For both, a change in x has a change in y no matter the value of

Estimations

  • The estimations we have used so far have assumed linear models
  • We often have a quantity x and we have to figure out m to get y.

Linear Fits

  • If we have a bunch of data that is roughly linear, we can extract a model

Unit Conversions

  • In a unit conversion we can plot the starting units on the x-axis as the dependent variable.
  • We can plot the ending units on the y-axis as the independent variable.
  • The “unit conversion” is the slope of this line.

Details

Intercepts

  • What types of models are likely to have an intercept?
  • What models will have an intercept of zero?

Linear Models

Name some models or phenomenon that exhibit a linear relationship.

  • Taxi cab ride
  • Pizza price with toppings
  • Electricity bill

Circle questions

  • Is the circumference of a circle is proportional to its radius?
  • Is the constant of proportionality the same for all circles?
  • Is the area of a circle proportional to its radius?

Lecture Notes