Chapter 2: Linear Functions
Chapter one was a window that gave us a peek into the entire course. Our goal was to understand the basic structure of functions and function notation, the toolkit functions, domain and range, how to recognize and understand composition and transformations of functions and how to understand and utilize inverse functions. With these basic components in hand we will further research the specific details and intricacies of each type of function in our toolkit and use them to model the world around us.
As we approach day to day life we often need to quantify the things around us, giving structure and numeric value to various situations. This ability to add structure enables us to make choices based on patterns we see that are weighted and systematic. With this structure in place we can model and even predict behavior to make decisions. Adding a numerical structure to a real world situation is called Mathematical Modeling.
When modeling real world scenarios, there are some common growth patterns that are regularly observed. We will devote this chapter and the rest of the book to the study of the functions used to model these growth patterns.
Section 2.1 Linear Functions
Section 2.2 Graphs of Linear Functions
Section 2.3 Modeling with Linear Functions
Section 2.4 Fitting Linear Models to Data
Section 2.5 Absolute Value Functions
Section 2.1 Linear Functions
As you hop into a taxicab in Las Vegas, the meter will immediately read $3.30; this is the “drop” charge made when the taximeter is activated. After that initial fee, the taximeter will add $2.40 for each mile the taxi drives. In this scenario, the total taxi fare depends upon the number of miles ridden in the taxi, and we can ask whether it is possible to model this type of scenario with a function. Using descriptive variables, we choose m for miles and C for Cost in dollars as a function of miles: C(m).
We know for certain that [pic], since the $3.30 drop charge is assessed regardless of how many miles are driven. Since $2.40 is added for each mile driven, then [pic]
If we then drove a second mile, another $2.40 would be added to the cost: [pic]
If we drove a third mile, another $2.40 would be added to the cost: [pic]
From this we might observe the pattern, and conclude that if m miles are driven,[pic] because we start with a $3.30 drop fee and then for each mile increase we add $2.40.
It is good to verify that the units make sense in this equation. The $3.30 drop charge is measured in dollars; the $2.40 charge is measured in dollars per mile. So [pic]
When dollars per mile are multiplied by a number of miles, the result is a number of dollars, matching the units on the 3.30, and matching the desired units for the C function.
Notice this equation [pic] consisted of two quantities. The first is the fixed $3.30 charge which does not change based on the value of the input. The second is the $2.40 dollars per mile value, which is a rate of change. In the equation this rate of change is multiplied by the input value.
Looking at this same problem in table format we can also see the cost changes by $2.40 for every 1 mile increase.
|m |0 |1 |2 |3 | |C(m) |3.30 |5.70 |8.10 |10.50 |
It is important here to note that in this equation, the rate of change is constant; over any interval, the rate of change is the same.
Graphing this equation, [pic] we see the shape is a line, which is how these functions get their name: linear functions
When the number of miles is zero the cost is $3.30, giving the point (0, 3.30) on the graph. This is the vertical or C(m) intercept. The graph is increasing in a straight line from left to right because for each mile the cost goes up by $2.40; this rate remains consistent.
In this example you have seen the taxicab cost...
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