Describe the transformations that produce the graph of g(x) = 1/2 (x-4)^3 +5 from the graph of the parent function f(x) = x^3 . Give the order in which they must be performed to obtain the correct graph.
To obtain the graph of g(x) = 1/2 (x-4)^3 +5 from the graph of f(x) = x^3, the following transformations need to be performed in the given order:
1. Horizontal translation: The graph of g(x) is shifted 4 units to the right from the graph of f(x) due to the term (x-4) in the equation. This horizontal translation is performed before any other transformation.
2. Vertical stretch/compression: The graph of g(x) is vertically compressed by a factor of 1/2 from the graph of f(x) due to the coefficient 1/2 multiplying the equation. This transformation is performed after the horizontal translation.
3. Vertical translation: The graph of g(x) is shifted 5 units up from the graph of f(x) due to the term +5 added to the equation. This transformation is performed after the vertical stretch/compression.
Therefore, the correct order of transformations is:
1. Horizontal translation by 4 units to the right.
2. Vertical stretch by a factor of 1/2.
3. Vertical translation by 5 units up.
To produce the graph of g(x) = 1/2 (x-4)^3 +5 from the parent function f(x) = x^3, we need to apply a series of transformations in the correct order. These transformations are:
1. Translation: Move the graph 4 units to the right (to the left if the sign is negative) to account for the x - 4 term in g(x). This means the graph of g(x) is shifted right by 4 units compared to f(x).
2. Vertical Scaling: Multiply the function by 1/2. This vertical scaling factor of 1/2 compresses the graph vertically. The function becomes narrower compared to f(x).
3. Vertical Translation: Add 5 to the function. This shifts the graph 5 units upward compared to f(x).
To obtain the correct graph, these transformations must be performed in the following order:
1. Vertical Scaling - Multiply the function by 1/2.
2. Translation - Move the graph 4 units to the right.
3. Vertical Translation - Add 5 to the function.
Performing the transformations in this order will allow you to obtain the correct graph of g(x) = 1/2 (x-4)^3 +5.
To describe the transformations that produce the graph of g(x) = 1/2 (x-4)^3 +5 from the parent function f(x) = x^3, we need to identify two key transformations: a vertical transformation and a horizontal transformation. Additionally, it is important to understand the order in which these transformations must be applied to obtain the correct graph.
1. Horizontal Transformation (Shift):
The function g(x) = 1/2 (x-4)^3 +5 involves a horizontal transformation. Specifically, we have a shift to the right by 4 units. This means that each x-coordinate in the original function f(x) = x^3 should be increased by 4 units to achieve the corresponding point on g(x).
2. Vertical Transformation (Stretch):
The function g(x) = 1/2 (x-4)^3 +5 involves a vertical transformation. It includes both a vertical stretch and a vertical shift. The parent function f(x) = x^3 is being stretched vertically by a factor of 1/2, which makes it half as tall. Additionally, there is a vertical shift upward by 5 units.
Now, let's discuss the order in which these transformations should be performed:
The order of transformations is crucial because performing them in a different order may yield a different graph. In this case, the correct order is as follows:
1. Perform the horizontal transformation (shift) first by moving each x-coordinate 4 units to the right. This gives us g(x) = f(x-4) = (x-4)^3.
2. Next, apply the vertical transformation. Start by stretching the graph of f(x-4) vertically by a factor of 1/2. This results in g(x) = 1/2 (x-4)^3.
3. Finally, add the vertical shift of 5 units, which gives us the final function: g(x) = 1/2 (x-4)^3 + 5.
By following this order of transformations, starting with the horizontal shift and then applying the vertical stretch and shift, we can obtain the correct graph of g(x) = 1/2 (x-4)^3 + 5 from the parent function f(x) = x^3.