V(r)= 4/3r^3
Do these transformations:
Horizontal shift to the right 2 units
Vertical dilation of 3
Vertical shift up 1 unit
To perform these transformations on the function V(r) = 4/3r^3:
1. Horizontal shift to the right 2 units:
We need to replace r with (r - 2) to achieve a horizontal shift to the right by 2 units.
V(r) = 4/3(r - 2)^3
2. Vertical dilation of 3:
We need to multiply the entire function by 3 to achieve a vertical dilation of 3.
V(r) = 3 * 4/3(r - 2)^3
Simplifying,
V(r) = 4(r - 2)^3
3. Vertical shift up 1 unit:
We need to add 1 to the function to achieve a vertical shift up by 1 unit.
V(r) = 4(r - 2)^3 + 1
Therefore, the transformed function is V(r) = 4(r - 2)^3 + 1.
To apply the given transformations to the equation V(r) = (4/3)r^3, follow these steps:
1. Horizontal shift to the right 2 units:
To shift the graph horizontally to the right, subtract the desired amount from the variable inside the equation.
V(r) = (4/3)(r - 2)^3
2. Vertical dilation of 3:
To vertically dilate the graph, multiply the equation by the desired factor.
V(r) = 3(4/3)(r - 2)^3
Simplifying, we get:
V(r) = 4(r - 2)^3
3. Vertical shift up 1 unit:
To shift the graph vertically, add the desired amount to the equation.
V(r) = 4(r - 2)^3 + 1
The final transformed equation is V(r) = 4(r - 2)^3 + 1.
To understand the transformations on the function V(r) = (4/3)r^3, let's break it down step by step.
1. Horizontal shift to the right 2 units:
To shift the graph horizontally, you need to replace the variable r with (r - h), where h is the amount of units you want to shift. In this case, the function becomes V(r - 2) = (4/3)(r - 2)^3. So, the original function has been shifted to the right 2 units.
2. Vertical dilation of 3:
To perform a vertical dilation, you need to multiply the entire function by a constant. In this case, we need to multiply V(r - 2) by 3. The resulting function would be 3V(r - 2) = 3(4/3)(r - 2)^3 = 4(r - 2)^3. So, the original function has been vertically dilated by a factor of 3.
3. Vertical shift up 1 unit:
To shift the graph vertically, you need to add a constant to the function. In this case, we need to add 1 to 3V(r - 2). The final transformed function would be 3V(r - 2) + 1 = 4(r - 2)^3 + 1. So, the original function has been shifted up 1 unit.
Therefore, the transformed function is V(r) = 4(r - 2)^3 + 1 after applying a horizontal shift to the right 2 units, a vertical dilation of 3, and a vertical shift up 1 unit.