Complete the table. Describe the combination of transformations step by step and write the equation that builds on the parent function, f(x)=x^3 from Figure 2A.1 to 2A.2.
The second graph has the points:
(3,4)
(2,1)
(1,-2)
please help me
To describe the combination of transformations step by step, let's analyze the given points:
(3,4) - This point tells us that the transformed graph has a point at (3,4), which means there is a vertical shift upwards by 4 units.
(2,1) - This point tells us that the transformed graph has a point at (2,1), which means there is a vertical shift upwards by 1 unit.
(1,-2) - This point tells us that the transformed graph has a point at (1,-2), which means there is a vertical shift downwards by 2 units.
Therefore, the combination of transformations step by step can be described as:
1. Start with the parent function f(x) = x^3.
2. Apply a vertical shift upwards by 4 units.
3. Apply a vertical shift upwards by another 1 unit.
4. Apply a vertical shift downwards by 2 units.
To write the equation that builds on the parent function, we can use the general equation for vertical shifts:
f(x) = a(x-h)^3 + k, where (h,k) represents the horizontal and vertical shifts.
Using the given information:
- First vertical shift upward by 4 units: f(x) = a(x-h)^3 + 4
- Second vertical shift upward by 1 unit: f(x) = a(x-h)^3 + 5
- Third vertical shift downward by 2 units: f(x) = a(x-h)^3 + 3
We do not have information about any horizontal shifts, so the equation becomes:
f(x) = a(x-h)^3 + 3
Since the parent function f(x) = x^3 has a vertex at (0,0), we can assume that h = 0, and the equation simplifies to:
f(x) = ax^3 + 3