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.

Question

The second graph has the points:
(3,4)
(2,1)
(1,-2)

Answer 1

To describe the combination of transformations step by step and write the equation that builds on the parent function from Figure 2A.1 to 2A.2, let's analyze the given points.

Figure 2A.1: Parent function f(x) = x^3

Let's first analyze the given points in Figure 2A.2: (3,4), (2,1), (1,-2).

1. Horizontal Translation:
The point (3,4) is one unit to the left of the origin. This indicates a horizontal translation to the right by 1 unit.

2. Vertical Translation:
The point (2,1) is translated 3 units vertically upwards from the origin. This represents a vertical translation upward by 3 units.

3. Reflection:
The point (1,-2) is reflected across the x-axis. This indicates a reflection of the function about the x-axis.

Combining all the transformations:
- Horizontal translation to the right by 1 unit. (x - 1)
- Vertical translation upward by 3 units. (x - 1) + 3
- Reflection across the x-axis. -((x - 1) + 3)

To build the equation that incorporates these transformations on the parent function f(x) = x^3, we have:

g(x) = -((x - 1) + 3)^3

Simplifying further,

g(x) = -(x - 1 + 3)^3

g(x) = -(x - 4)^3

So, the equation that builds on the parent function f(x) = x^3 to get the graph in Figure 2A.2 is g(x) = -(x - 4)^3.