State the parent function that must be transformed to create the graph of y=2/3x^3 - 2. Describe the transformation that must be applied to the parent function.
Parent function= x^3
I don't know the transformation...
scale y by 2/3 and then shift downward 2 units.
if f = x^3
and g-k = a(x-h)^3
shift the graph h units to the right
scale y by a factor of a
shift the graph up by k units
To create the graph of y = (2/3)x^3 - 2, we start with the parent function y = x^3 and apply a vertical stretch, a vertical shift, and a vertical reflection.
1. Vertical Stretch: The coefficient 2/3 indicates a vertical stretch of the parent function. The value is between 0 and 1, so the graph is compressed vertically. The stretch factor of 2/3 means that the y-values are scaled by a factor of 2/3 compared to the parent function.
2. Vertical Shift: The constant term -2 indicates a vertical shift downward by 2 units. The graph is moved downward without changing its shape.
3. Vertical Reflection: The coefficient in front of the x^3 term is positive, so there is no vertical reflection.
In summary, to create the graph of y = (2/3)x^3 - 2, we vertically stretch the parent function y = x^3 by a factor of 2/3, shift it downward by 2 units, and there is no vertical reflection.
To determine the parent function that must be transformed to create the graph of y = 2/3x^3 - 2, we need to examine the given equation. In this case, the parent function is x^3.
The transformation applied to the parent function involves two main operations:
1. Horizontal Stretch/Compression: The coefficient in front of the x^3 term, which is 2/3, determines the horizontal scaling factor. If the coefficient is greater than 1, it indicates a horizontal compression, whereas if it is less than 1, it implies a horizontal stretch. In this case, the coefficient is 2/3, which is less than 1, so it represents a horizontal stretch.
2. Vertical Shift: The constant term, -2, indicates a vertical shift of the graph. If the constant term is positive, the graph is shifted upward, and if negative, it is shifted downward. In this case, the constant term is -2, which implies a vertical shift of 2 units downward.
By applying these transformations to the parent function x^3, we can obtain the graph of y = 2/3x^3 - 2.