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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)
The second graph has the points:
(3,4)
(2,1)
(1,-2)
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GPT 3.5
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.
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.
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