The Value of E – Euler Functions and Transformation of Functions

The Euler Functions
The Euler Functions

We are going to continue examining some essential pre calculus functions. Euler’s constant (e) is a very important number in calculus. The value of e is equal to approximately 2.71828 and is the most convenient base for dealing with derivatives.

The reason for this comes down to limit analysis of a derivative (more on this in the limits section). Essentially though, it comes from the fact that if you take the derivative of an exponential function with base e (a Euler function), then the multiplicative constant of the derivative is just 1. It is important that you have a graphing calculator that is accurate and fast.

Graphically this is seen by the slope of the tangent at x=0 is one. Do not worry too much about this yet, we will see the full details of why this is important in the near future.We see in this lesson that e^x (the euler function) is between 2^x and 3^x and we see that the slope of then tangent at the y-axis is exactly 1.

The example in this video asks us to plot the graph of y = e^-x +2 , and to find its domain and range. We demonstrate this by plotting some points. By doing so we see that the graph is defined for all values of x, but y must be greater than 2.

The exponential function with base ‘e’ does not produce any negative values. Its lowest value is 0, and since we have a constant vertical translation of 2, the lowest value of the entire function will be 2.Even though you will not use the exact value of e very often, it is important to know where it stand numerically.

This is especially true when it comes time to plot exponential functions with different base values. Overall, euler’s number is one of the most convenient bases for studying calculus. This will become much more apparent when we get into differentiation rules and limits.

Transformation of Functions

The transformation of functions.
The transformation of functions.

Also, today we will be discussing the transformation of functions. One of the last algebra skill sets needed for differential studies. Function transformations are useful for graphing by hand. If we see that a complicated function is just a transformation of an easy function, then it is much easier to graph.

We see an example of this with a simple parabola y=x^2.Function translations are the easiest form of transformation. We see that we can move a function vertically by simply adding or subtracting a constant.

The graph y=x^2 can be shifted up 2 units; y=x^2+2. Horizontal translations are similar except that they move the original function left or right. Horizontal translations add or subtract a constant to the argument of the function. For example the graph of y=x^2 can be shifted to the right by 4 units; y=(x-4)^2.

Function stretching is achieved by multiplying a constant into our function. Similar to translations if the constant is applied to the whole function, then it will stretch vertically. If the constant is attached to the variable (x) then it will stretch in the horizontal direction.

I hope you learn some knowledge kids. Remember to study hard and math will become easy as 1,2,3! Good luck students!