Linear Functions

Linear Functions Introduction

Many of our questions and estimations can be determined by linear relationships.

A linear relationship means that 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.

Concepts

• Independent variable

• Dependent variable

• Slope

• Proportional Relationship

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

• Has a linear relationship

• The independent variable is zero when the dependent variable is zero

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 x.

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 Extrapolation

• If we assume a relationship, we can predict its value in the future

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

Proportion

• If $b=0$ we say $x$ and $y$ are proportional
• Symbol $\propto$
• If two things are proportional, the values of the two properties are related by a constant factor

• For the purposes of estimation, we may treat certain things as proportional as an approximation.

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?

Linear relationships

• What other quantities have linear relationships?

When does proportion and linearity stop being useful?

• What is linear?

• What is not linear?

Activities

What is well modeled by linear functions?

What questions require linear models to answer?

• Brainstorm a list of questions you are interested where the answer

  will be a slope, an intercept, or a linear projection.

What are examples of linear fits?

What have we already covered that are linear relationships?

• Unit conversion

• Estimations