This course is an investigation into the Elementary functions. One of our goals is simply to build a library of the Elementary functions, their properties, and how we represent them. From there, we can build more complex functions from these building blocks and compare them to each other.

Our first exploration is into linear functions. The distinguishing characteristic about linear functions is that they have a constant growth rate or constant rate of change.

Linear functions relate two measurements, which change proportionally. That is when one measurement changes, the other changes by a constant multiple of the first change. This constant multiple acts as a conversion factor. We call it a rate.

This rate is constant throughout a linear function.

In other words, if you calculate the rate of change from any pair in a linear function to any other pair, you always get the same result.

Let $f$ be the linear function. Let $(a, f(a))$ and $(b, f(b))$ any two pairs in the function.

The rate of change from $a$ to $b$ is

$$\frac {f(b) - f(a)}{b-a}$$

For a linear function, this calculation gives the same result, no matter which two distinct pairs you choose.

From this we discover that linear functions are those functions that can be described as

$$L(d) = A \cdot d + B$$

where $A$ and $B$ are real numbers.

The graphs of linear functions are lines and we would like to be able to work back and forth between the algebra and geometry.

Learning Outcomes

In this section, students will

• examine constant growth rates.
• produce graphs of linear functions.
• produce formulas from graphs.
• create tangent lines.
• solve systems of linear equations.