Transforming Exponential Equations

Step 1: Identify the parent function
Step 2: Use a graphing calculator to graph and compare to the parent function
Step 3: Observe and draw conclusions on the effect each operation has on the graph of an exponential equation
Step 4: Use those observations to predict what will happen to a graph given an exponential equation

Step 1: Parent Functions

A parent function is the simplist form of an equation in a family of functions. For example, in the linear equation family y=mx+b is most common form of an equation so the simplist form would be y=x.

Exponential Parent Function
The standard from of an exponential function y=a(b)^(x-h)+k. This leaves the parent function as y=b^x. For the purpuse of this assignment y=2^x will be the parent function that every other function will be compared to.

Step 2: Graphing Calculator

The graphing calculator that will be used here is a free online graphing calculator from
In order to graph an exponent you will need to be able to type an exponent into this calculator. Look for this button:
To write more than just an x in the exponent like y=2^(x+5) you will need to use parentheses (). If you type the first parentheses before you type anything else it will insert the ending parentheses looking like this: 2. Then you will be able to write inside the parentheses.

Step 3: Determining the Transformation.

The table below contains many equations to graph and compare to the parent fucntion. These are just the basics of discovering transformations of exponential equations. This table only explores integer operations. Your job is to explore what each small change to the parent function does. In doing so you are looking for patterns that will help you in more complex situations. The first one has been done as an example -- the amount of change does matter. Words you should be using to describe the change are slides, expands, shrinks, flips, or rotates.

y=2^x	PARENT FUNCTION
Equation	Change	Amount of Change
Addition
y=2^x+1	The graph slides up	Up 1 unit
y=2^x+2
y=2^x+(-1)
y=2^x+(-2)
Subtraction
y=2^x-1
y=2^x-2
y=2^x-(-1)
y=2^x-(-2)
Multiplication
y=2(2)^x
y=3(b)^x
y=-2(2)^x
y=-3(2)^x
Division
y=(2^x)/2
y=(2^x)/3
y=(2^x)/(-2)
y=(2^x)/(-3)
Addition	in the Exponent
y=2^(x+1)
y=2^(x+2)
y=2^(x+(-1))
y=2^(x+(-2))
Subtraction	in the Exponent
y=2^(x-1)
y=2^(x-2)
y=2^(x-(-1))
y=2^(x-(-2))
Multiplication	in the Exponent
y=2^(2x)
y=2^(3x)
y=2^(-2x)
y=2^(-3x)
Division	in the Exponent
y=2^(x/2)
y=2^(x/3)
y=2^(x/(-2))
y=2^(x/(-3))
Changing	the Base Number
y=3^x
y=4^x

You may have noticed that we did not change the base to a negative integer. That is because a negative base creates a non-continous graph, called a discreet graph (none of the points are connected). The space between the points is connected by what we call imaginary numbers. However, imaginary numbers are outside the scope of this lesson which as stated above is only dealing with integers.

Step 4: Predicting the Transformation

Now that all of your observations have been made and your conclusions have been drawn, let's see how well your conclusions hold up when you are to use them as predictions. Perdict what you think will happen in the following graphs.
(1) y=2^x+10
(2) y=2^(x-5)
(3) y=7(2)^x
(4) y=2^(x/(-11))
(5) y=10^x
(6) y=4(2)^(x-5)+7 (Can you get all three for this one?)