graph the equation shown below by transforming the given graph of the parent function. \n$y = \\frac{1}{8}…

Explanation:

Step1: Identify the parent function

The parent function here is ( y = 2^x ), which is an exponential function with a base of 2. The given graph in the image appears to be the graph of ( y = 2^x ) (passing through points like (0,1), (1,2), (2,4), (3,8) as seen from the blue dots: when ( x = 0 ), ( y = 1 ); ( x = 1 ), ( y = 2 ); ( x = 2 ), ( y = 4 ); ( x = 3 ), ( y = 8 )).

Step2: Analyze the transformation

The given equation is ( y=\frac{1}{8}\cdot2^x ). We can rewrite ( \frac{1}{8} ) as ( 2^{-3} ) because ( 2^3 = 8 ), so ( \frac{1}{8}=2^{-3} ). Then the equation becomes ( y = 2^{-3}\cdot2^x ). Using the exponent rule ( a^m\cdot a^n=a^{m + n} ), we get ( y = 2^{x-3} ). This represents a horizontal shift of the parent function ( y = 2^x ) to the right by 3 units, or alternatively, a vertical compression by a factor of ( \frac{1}{8} ) (since multiplying the function by ( \frac{1}{8} ) compresses it vertically).

To graph ( y=\frac{1}{8}\cdot2^x ) from ( y = 2^x ):

• For the parent function ( y = 2^x ), when ( x = 0 ), ( y = 1 ); ( x = 1 ), ( y = 2 ); ( x = 2 ), ( y = 4 ); ( x = 3 ), ( y = 8 ).
• For the transformed function ( y=\frac{1}{8}\cdot2^x ), we multiply the ( y )-values of the parent function by ( \frac{1}{8} ):
  • When ( x = 0 ), ( y=\frac{1}{8}\cdot2^0=\frac{1}{8}\cdot1=\frac{1}{8} )
  • When ( x = 1 ), ( y=\frac{1}{8}\cdot2^1=\frac{1}{8}\cdot2=\frac{1}{4} )
  • When ( x = 2 ), ( y=\frac{1}{8}\cdot2^2=\frac{1}{8}\cdot4=\frac{1}{2} )
  • When ( x = 3 ), ( y=\frac{1}{8}\cdot2^3=\frac{1}{8}\cdot8 = 1 )
  • When ( x = 4 ), ( y=\frac{1}{8}\cdot2^4=\frac{1}{8}\cdot16 = 2 )
  • When ( x = 5 ), ( y=\frac{1}{8}\cdot2^5=\frac{1}{8}\cdot32 = 4 )
  • When ( x = 6 ), ( y=\frac{1}{8}\cdot2^6=\frac{1}{8}\cdot64 = 8 )

So the key points of the transformed function will be ( (0,\frac{1}{8}) ), ( (1,\frac{1}{4}) ), ( (2,\frac{1}{2}) ), ( (3,1) ), ( (4,2) ), ( (5,4) ), ( (6,8) ), etc. This is equivalent to shifting the parent graph ( y = 2^x ) (which has points (0,1), (1,2), (2,4), (3,8)) to the right by 3 units (since when ( x = 3 ) in the transformed function, ( y = 1 ), which is the ( y )-value of the parent function at ( x = 0 ); when ( x = 4 ), ( y = 2 ), which is the parent's ( y )-value at ( x = 1 ), etc.).

Answer:

To graph ( y=\frac{1}{8}\cdot2^x ), we start with the parent function ( y = 2^x ) (exponential growth, passing through (0,1), (1,2), (2,4), (3,8)). Then we apply a vertical compression by a factor of ( \frac{1}{8} ) (or a horizontal shift right by 3 units). The transformed graph will pass through points like ( (0,\frac{1}{8}) ), ( (1,\frac{1}{4}) ), ( (2,\frac{1}{2}) ), ( (3,1) ), ( (4,2) ), etc., and will have the same shape as ( y = 2^x ) but compressed vertically (or shifted horizontally) to match the equation ( y=\frac{1}{8}\cdot2^x ).