graph the equation shown below by transforming the given graph of the parent function. \n$y = 2x^3$

graph the equation shown below by transforming the given graph of the parent function. \n$y = 2x^3$

graph the equation shown below by transforming the given graph of the parent function. \n$y = 2x^3$

Answer

Explanation:

Step1: Identify the parent function

The parent function for ( y = 2x^3 ) is ( y = x^3 ). The graph of ( y = x^3 ) is a cubic curve passing through the origin, symmetric about the origin, and has a point - shape that increases from left to right (for positive ( x )) and decreases from left to right (for negative ( x )) with a relatively "flat" area near the origin and steeper as ( |x| ) increases.

Step2: Analyze the transformation (vertical stretch)

The equation ( y = 2x^3 ) represents a vertical stretch of the parent function ( y=x^3 ) by a factor of 2. For a function ( y = a\cdot f(x) ), when ( |a|> 1 ), it is a vertical stretch of the graph of ( y = f(x) ) by a factor of ( |a| ).

To graph ( y = 2x^3 ) from ( y=x^3 ):

  • For a given ( x ) - value, the ( y ) - value of ( y = 2x^3 ) is twice the ( y ) - value of ( y=x^3 ).

  • Let's take some key points of the parent function ( y = x^3 ):

    • When ( x = 0 ), ( y=x^3=0 ), and for ( y = 2x^3 ), ( y = 2\times0^3=0 ), so the point ( (0,0) ) remains the same.
    • When ( x = 1 ), ( y=x^3 = 1 ), and for ( y = 2x^3 ), ( y=2\times1^3 = 2 ), so the point ( (1,1) ) on ( y = x^3 ) becomes ( (1,2) ) on ( y = 2x^3 ).
    • When ( x=- 1 ), ( y=x^3=-1 ), and for ( y = 2x^3 ), ( y=2\times(-1)^3=-2 ), so the point ( (-1, - 1) ) on ( y = x^3 ) becomes ( (-1,-2) ) on ( y = 2x^3 ).
    • When ( x = 2 ), ( y=x^3 = 8 ), and for ( y = 2x^3 ), ( y=2\times2^3=16 ), so the point ( (2,8) ) on ( y = x^3 ) becomes ( (2,16) ) on ( y = 2x^3 ).
    • When ( x=-2 ), ( y=x^3=-8 ), and for ( y = 2x^3 ), ( y=2\times(-2)^3=-16 ), so the point ( (-2,-8) ) on ( y = x^3 ) becomes ( (-2,-16) ) on ( y = 2x^3 ).

  • Plot these new points ( (0,0) ), ( (1,2) ), ( (-1,-2) ), ( (2,16) ), ( (-2,-16) ) and draw a smooth curve through them. The shape of the curve will be similar to the parent cubic curve ( y = x^3 ), but it will be "taller" (vertically stretched) because the ( y ) - values are multiplied by 2. The curve will still pass through the origin, be symmetric about the origin, and have the general cubic curve shape with a steeper slope (for the same change in ( x )) compared to the parent function ( y = x^3 ).

Answer:

To graph ( y = 2x^3 ):

  1. Start with the parent function ( y=x^3 ), which has a cubic curve shape passing through the origin, symmetric about the origin.
  2. Apply a vertical stretch by a factor of 2: For each ( x ), calculate ( y = 2x^3 ) (double the ( y ) - value of ( y=x^3 ) at that ( x )).
  3. Plot key points like ( (0,0) ), ( (1,2) ), ( (-1,-2) ), ( (2,16) ), ( (-2,-16) ) and draw a smooth cubic - shaped curve through them. The graph will be a vertically stretched (by factor 2) version of the parent cubic function ( y = x^3 ), passing through the origin, symmetric about the origin, and steeper than ( y=x^3 ) for the same ( |x|>0 ).