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