I don't understand the answer:
Identify the transformations of the graph of ƒ (x) = x 3
that produce the graph of the
given function g (x). Then graph g (x) on the same coordinate plane as the graph of ƒ (x)
by applying the transformations to the reference points (-1, -1), (0, 0) , and (1, 1)
g (x) = - 1/2(x-3)^3
answer:
The transformations of the graph of f (x)
that produce the graph of g (x) are:
• a vertical compression by a factor of 1/2
• a reflection across the x-axis
• a translation of 3 units to the right
thank you again
Let's break down the transformations step-by-step:
1. Vertical Compression by a factor of 1/2:
The original function f(x) = x^3 has a vertical compression by a factor of 1/2, which means the y-values will be halved. This transformation affects the vertical scaling of the graph.
2. Reflection across the x-axis:
Next, there is a reflection across the x-axis. This means that the graph of f(x) will be reflected so that all the points on the graph that were above the x-axis will now be below the x-axis, and vice versa. This transformation affects the orientation of the graph.
3. Translation 3 units to the right:
Finally, there is a translation of 3 units to the right. This means that all the x-values of the graph will be increased by 3, shifting the graph to the right. This transformation affects the position of the graph horizontally.
To graph g(x) = -1/2(x-3)^3 on the same coordinate plane as f(x), you can apply these transformations to the reference points (-1, -1), (0, 0), and (1, 1).
For example, for the first reference point (-1, -1):
1. Apply vertical compression: The y-value becomes -1/2 times the original value, so it becomes -1/2*(-1) = 1/2.
2. Apply reflection across x-axis: The y-value changes sign, so it becomes -1/2.
3. Apply translation 3 units to the right: The x-value increases by 3, so it becomes -1 + 3 = 2.
So the transformed point is (2, -1/2).
Similarly, you can apply these transformations to the other reference points (0, 0) and (1, 1) to find their corresponding transformed points.
Once you have the transformed points, plot them on the same coordinate plane as the graph of f(x), connecting the points to obtain the graph of g(x).
To understand the transformations that produce the graph of the function g(x) = -1/2(x-3)^3 from the graph of the function f(x) = x^3, we'll break them down step by step:
1. Vertical Compression by a factor of 1/2:
Start with the graph of f(x) = x^3. To vertically compress the graph, multiply the y-values of each point on the graph by 1/2. This makes the graph narrower compared to the original.
2. Reflection across the x-axis:
Reflect the vertically compressed graph from step 1 across the x-axis. This means that the positive y-values become negative, and the negative y-values become positive. This transformation flips the graph upside down.
3. Translation of 3 units to the right:
Once you have the reflected graph, shift it horizontally to the right by 3 units. This means that every point on the graph moves 3 units to the right. The translated graph will now be positioned 3 units to the right of the original graph.
By applying these three transformations to the reference points (-1, -1), (0, 0), and (1, 1) of the original function f(x) = x^3, you will obtain the corresponding points for the transformed function g(x) = -1/2(x-3)^3.
To graph g(x) on the same coordinate plane as f(x), plot the transformed points and connect them to visualize the graph of g(x).