Exponential functions and logarithms: Exponential functions
Transformations of the exponential function
We have previously looked at the graph of the function #\blue{a}^x#. Using three transformations, we can reshape this graph. For every transformation, we will give the corresponding asymptote, domain and range. Remember that of the regular exponential function #\blue{a}^x# the asymptote is the line #y=0#, the domain all numbers, and the range only the positive numbers.
Transformations
Below we can see the three transformations of the function #y=\blue{a}^x#. In all three of the images, the dotted line is the graph of the function #\blue{a}^x#.
Transformations  Examples  
1 
We shift the graph of #\blue{a}^x# upwards by #\green{b}#. The new formula becomes \[y=\blue{a}^x+\green b\] The line #y=\green{b}# becomes the asymptote, the domain is still all the numbers, and the range all numbers greater than #\green{b}#. 
geogebra plaatje

2 
We shift the graph of #y=\blue{a}^x# to the right by #\purple{c}#. The new formula becomes \[y=\blue{a}^{x\purple{c}}\] The asymptote, the domain and the range stay the same. 
geogebra plaatje

3 
We multiply the graph of #y=\blue{a}^x# by #\orange{d}# relative to the #y#axis. The new formula becomes \[y=\blue{a}^{\frac{1}{\orange{d}}\cdot x}\] The asymptote, the domain and the range stay the same. 
geogebra plaatje

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