graph the equation shown below by transforming the given graph of the parent function. \n$y = 4 \\cdot 2^x$

graph the equation shown below by transforming the given graph of the parent function. \n$y = 4 \\cdot 2^x$

graph the equation shown below by transforming the given graph of the parent function. \n$y = 4 \\cdot 2^x$

Answer

Explanation:

Step1: Identify the parent function

The parent function here is ( y = 2^x ). The given graph (with points like (0,1), (1,2), (2,4), (3,8) maybe? Wait, the green dot at (0,1), then blue dots: let's check the parent function ( y = 2^x ): when ( x = 0 ), ( y = 1 ); ( x = 1 ), ( y = 2 ); ( x = 2 ), ( y = 4 ); ( x = 3 ), ( y = 8 ). Wait, the given graph has a green dot at (0,1), then blue dots: maybe (1,2), (2,4), (3,8)? Wait, the equation is ( y = 4 \cdot 2^x ), which is a vertical stretch of the parent function ( y = 2^x ) by a factor of 4.

Step2: Recall vertical stretch transformation

For a function ( y = a \cdot f(x) ), if ( |a| > 1 ), it's a vertical stretch by a factor of ( |a| ). Here, ( a = 4 ), so we stretch the parent function ( y = 2^x ) vertically by 4.

Step3: Apply the transformation to key points

  • Parent function ( y = 2^x ) has key points: (0, 1), (1, 2), (2, 4), (3, 8), etc.
  • For ( y = 4 \cdot 2^x ), multiply the y - coordinates by 4:
    • When ( x = 0 ): ( y = 4 \cdot 2^0 = 4 \cdot 1 = 4 )
    • When ( x = 1 ): ( y = 4 \cdot 2^1 = 4 \cdot 2 = 8 )
    • When ( x = 2 ): ( y = 4 \cdot 2^2 = 4 \cdot 4 = 16 ) (but the graph grid goes up to y = 10, maybe we use smaller x or check the original graph's points. Wait, the original graph (parent) has a green dot at (0,1), blue dots at (1,2), (2,4), (3,8)? Wait, no, in the given graph, the green dot is at (0,1), then a blue dot at (1,2), (2,4), (3,8)? Wait, the y - axis goes up to 10. Wait, maybe the parent function is ( y = 2^x ), and we need to transform it to ( y = 4 \cdot 2^x ). So the transformation is a vertical stretch by 4. So the new points will be:
      • (0, 4) [since 4*1 = 4]
      • (1, 8) [since 4*2 = 8]
      • (2, 16) but 16 is above the grid, maybe ( - 1, 2) [x=-1: ( y = 4 \cdot 2^{-1}=4*(1/2)=2 )]
      • ( - 2, 1) [x=-2: ( y = 4 \cdot 2^{-2}=4*(1/4)=1 )]

Step4: Graph the transformed function

Plot the new points: ( - 2, 1), ( - 1, 2), (0, 4), (1, 8), etc. The graph will have the same shape as the parent exponential function ( y = 2^x ) but stretched vertically by a factor of 4, so it will be steeper and the y - values will be 4 times larger than the parent function at each x.

Answer:

To graph ( y = 4 \cdot 2^x ) from the parent function ( y = 2^x ):

  1. Identify Transformation Type: ( y = 4 \cdot 2^x ) is a vertical stretch of ( y = 2^x ) by a factor of 4 (since ( y = a\cdot f(x) ) with ( a = 4>1 ) implies vertical stretch).

  2. Transform Key Points:

    • Parent points for ( y = 2^x ): ((x, 2^x)) (e.g., ((0,1)), ((1,2)), ((2,4)), ((-1, 0.5)), ((-2, 0.25))).
    • For ( y = 4\cdot 2^x ), new points: ((x, 4\cdot 2^x)) (e.g., ((0,4)), ((1,8)), ((2,16)), ((-1,2)), ((-2,1))).

  3. Plot and Draw: Plot the transformed points (e.g., ((-2,1)), ((-1,2)), ((0,4)), ((1,8))) and draw the curve, which retains the exponential shape but is steeper (vertical stretch) compared to ( y = 2^x ).

(Note: The actual graphing involves plotting these points on the coordinate plane and drawing a smooth exponential curve through them, with the y - intercept at (0,4) and passing through (1,8), etc.)