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.
To describe the combination of transformations step by step from Figure 2A.1 to 2A.2 and find the equation that builds on the parent function f(x) = x^3, we will need more information about the specific transformations and the appearance of the figures. Without this information, we cannot provide a step-by-step analysis or write the equation accurately. Could you please provide more details about the transformations and/or the figures?
Unfortunately, I am unable to observe the given figures and provide any specific description or equation. However, I can explain the general approach to describing a combination of transformations step by step.
To describe the combination of transformations, we typically consider the order in which the transformations occur. The common transformations include translations (shifting the graph), reflections (flipping the graph), stretches/compressions (changing the size/scale of the graph), and vertical shifts (moving the entire graph upward or downward).
Given the parent function f(x) = x^3, let's suppose the combination of transformations from Figure 2A.1 to 2A.2 involves the following steps:
Step 1: Translation
The graph is translated horizontally and/or vertically. Let's assume it is translated horizontally to the right by 2 units.
Step 2: Stretch
The graph is stretched or compressed either horizontally or vertically. Let's assume it is stretched vertically by a factor of 2.
Step 3: Vertical Shift
The graph is shifted vertically. Let's assume it is shifted upward by 3 units.
Combining all the transformations, the equation that builds on the parent function, f(x) = x^3, could be:
y = 2(x - 2)^3 + 3
However, please note that this is a general example, and the actual transformation and resulting equation may differ depending on the specific figures provided in 2A.1 and 2A.2.
To complete the table and describe the combination of transformations, we need to understand the different types of transformations and how they affect a function.
The four main types of transformations are:
1. Translation: This transformation moves the entire graph of a function horizontally or vertically.
2. Reflection: This transformation flips the graph of a function over a line.
3. Stretch/Compression: This transformation stretches or compresses the graph of a function horizontally or vertically.
4. Vertical Shift: This transformation changes the position of the graph of a function vertically.
Now, let's examine the given parent function, f(x) = x^3, and the table to find the combination of transformations.
Figure 2A.1:
|x |f(x) |
|---|-----|
|0 |0 |
|1 |1 |
|-1 |-1 |
|2 |8 |
|-2 |-8 |
From Figure 2A.1 to Figure 2A.2, we can see there are multiple transformations occurring. Here's a step-by-step breakdown:
1. Translation: The graph has shifted horizontally and vertically. To determine the horizontal shift, we compare the x-values in the table. In Figure 2A.1, we have x = 0, 1, -1, 2, -2. In Figure 2A.2, the x-values are 1, 2, -2, 3, -3, respectively. The x-values have shifted by 1 to the right. The vertical shift can be determined by comparing the corresponding y-values for each x-value. In Figure 2A.1, we have f(0) = 0, f(1) = 1, f(-1) = -1, f(2) = 8, f(-2) = -8. In Figure 2A.2, the y-values are f(1) = 5, f(2) = 4, f(-2) = -6, f(3) = 27, f(-3) = -24. The y-values have shifted by 5 units upward.
2. Stretch/Compression: The graph of f(x) = x^3 has been stretched or compressed. To determine the stretch or compression factor, we compare the changing y-values for each x-value in both figures. In Figure 2A.1, the y-values change as follows: 0, 1, -1, 8, -8. In Figure 2A.2, the corresponding y-values are 5, 4, -6, 27, -24. By comparing the values, we can see that the y-values in Figure 2A.2 have been multiplied by a factor of 5/8.
3. Vertical Shift: The graph has been shifted vertically. We determined this in the first step when we observed the y-values shifting upward by 5 units.
Now, let's write the equation that builds on the parent function, f(x) = x^3, for Figure 2A.2. We'll use the step-by-step transformations we identified:
1. Start with the parent function, f(x) = x^3.
2. Apply the horizontal shift of 1 unit to the right: f(x) = (x - 1)^3. (Translation)
3. Apply the vertical shift of 5 units upward: f(x) = (x - 1)^3 + 5. (Vertical Shift)
4. Apply the vertical stretch/compression factor of 5/8: f(x) = 5/8 * (x - 1)^3 + 5. (Stretch/Compression)
Therefore, the equation that builds on the parent function, f(x) = x^3, for Figure 2A.2 is f(x) = 5/8 * (x - 1)^3 + 5.