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)
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.
To complete the table and describe the combination of transformations step by step, we need to understand the changes that have occurred from the parent function, f(x) = x^3, to the second graph in Figure 2A.2.
First, let's analyze the given points (3,4), (2,1), and (1,-2):
(3,4):
The x-coordinate has remained the same, but the y-coordinate has changed from 27 (as it would be in the parent function) to 4. This indicates a vertical compression since the y-values have become smaller.
(2,1):
Similarly, the x-coordinate has remained the same, but the y-coordinate has changed from 8 to 1. Again, this suggests a vertical compression.
(1,-2):
Once more, the x-coordinate is unchanged, but the y-coordinate has changed from 1 to -2. Again, there is a vertical compression going on.
Based on these observations, it appears that the given graph has undergone a vertical compression, causing the y-values to become smaller.
To construct the equation describing the transformation, we can consider the general equation for vertical compression:
f(x) = c * g(x)
Where c is the compression factor and g(x) is the parent function.
Since the y-values have become smaller, we know that the compression factor c is less than 1. Let's calculate it by comparing any corresponding y-values between the parent function and the second graph. Let's use the (2, 1) point:
c = (y-value in second graph) / (y-value in parent function)
c = 1 / 8
c = 1/8
Therefore, the equation for the transformation from the parent function (f(x) = x^3) to the second graph is:
f(x) = (1/8) * x^3
To determine 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, we can analyze the given points and their corresponding transformations.
Let's start with the given parent function f(x) = x^3.
Step 1: Vertical Translation:
The point (3, 4) suggests that Figure 2A.2 has been moved upward by 4 units. To achieve this, we subtract 4 from the original function.
f(x) = x^3 - 4
Step 2: Horizontal Translation:
The point (2, 1) indicates that Figure 2A.2 has been shifted 2 units to the left. To accomplish this, we add 2 to the x-value inside the function.
f(x + 2) = (x + 2)^3 - 4
Step 3: Vertical Scaling:
The point (1, -2) suggests that Figure 2A.2 has been vertically compressed by a factor of 2. To accomplish this, we multiply the function by 1/2.
f(x + 2) = (1/2)(x + 2)^3 - 4
Therefore, the equation that builds on the parent function f(x) = x^3 for the transformation from Figure 2A.1 to 2A.2 is f(x + 2) = (1/2)(x + 2)^3 - 4.