Monday, December 21, 2009

6.1 The Exponential Function and its Inverse

This is our second last unit of our Advanced Functions course. YAYYYY ^^
In the lesson, it will be separated into two parts. The first part will talk about the exponential functions and the second part will talk about its inverse.

Since this is the first lesson of the new logarithmic unit, most of the information presented should be learned by students.

PART 1- EXPONENTIAL FUNCTIONS

In part 1, we will learn how to solve for the exponent with different bases. The "log" function on your calculators will be used greatly in this unit.

Side note: the default base of the "log" key on your calculators is ALWAYS 10.

PART 2- Inverse

In part 2, we will be pretty much doing the same thing as grade 11. Switch x and y in the same equation and isolate for X again.

Exponential functions can be represented in 3 DIFFERENT ways.

1: General Form- the simplest form of representation of an exponential equation

y=5x or y=1/5x

2. Table of Values- a table which a list of the "x" values with the corresponding "y" values

REMEMBER: the first ratio of differences of the x and y values are always constant.

How do we find the first ratio of differences?

The values of : Y2-Y1 = answer.

3: Exponential functions- this is a graph representation of the exponential functions. However, just by looking at a graph cannot help you determine whether an exponential function is or not.

Rate of Change:

To determine the rage of change of an exponential function is really much similar to finding the rate of change of any other functions.

1) Pick the x values that are really close to the value you are given to find the IROC.

2) For example: if the value that is given is 2. You may go to the left of the value or right to the value which results in x = 2.0001 or x = 1.9999.

3) Sub these values back into the original equation and find the “y” values

4) To choose the two closet points that lead to the given value will give you the best and most accurate slope of secant that is required for the answer.

5) After having the “x” values and the “y” values. Put it into the general equation used to find slopes which is:

AROC [x value – x value] = y2-y1 / x2-x1

6) To conclude your answer, you must write, “Therefore, the IROC of the given value is approximately at “answer of AROC of the 2 x values”.

Inverse functions:

The graph of the inverse function of a logarithmic function is a reflection on the y= x function.

To find an inverse function of a logarithmic function, switch x and y from the equation and solve for x again.

By switching (x,y) -> (y,x) , these can be your new inverse values that can be used to apply transformations.

Differences and similarities between an inverse function and an exponential function

1) A exponential function will have an increasing slope while the inverse function will have an decreasing slope

2) A horizontal asymptote (y = a value) is present in an exponential function. A vertical asymptote (x = a value) is present in the inverse function

3) Depending on the transformations, the horizontal asymptote or the vertical asymptote is affected by the vertical or horizontal shifts.