Linear Scales

When we are reading most graphs and maps, there is a linear, proportional relationship between the distance on the page or screen and the data.

  • On a map, there is a bar showing the scale length of a mile and a kilometer.
  • On a graph, there is a relationship between the length on the page and the data.

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