how was the graph g is obtained from the graph of f.
4. f(x) = x^3 ; g(x) = (x - 1)^3 - 1
you are looking at a simple translation
f(x) was moved to the right one unit, and moved down one unit
for each function, find a few points, then plot each curve to confirm
To obtain the graph of g(x) from the graph of f(x), we need to apply two transformations: a horizontal shift and a vertical shift.
1. Horizontal shift: The graph of g(x) is obtained by shifting the graph of f(x) one unit to the right. Since g(x) = (x - 1)^3 - 1, every x-coordinate of f(x) is decreased by 1 to get the corresponding x-coordinate of g(x). This means that g(x) will have the same shape as f(x), but shifted one unit to the right.
2. Vertical shift: After the horizontal shift, we subtract 1 from the y-coordinate for each point on the shifted graph. This vertical shift downward by 1 unit moves the entire graph of g(x) down by 1 unit compared to f(x).
Therefore, the graph of g(x) is obtained by shifting the graph of f(x) one unit to the right and one unit down.
To obtain the graph of function g from the graph of function f, we need to apply two transformations: a horizontal shift and a vertical shift.
1. Horizontal shift:
The term (x - 1) in the function g(x) = (x - 1)^3 - 1 indicates a horizontal shift of one unit to the right. This means that every x-coordinate of the graph of f(x) needs to be increased by one to obtain the corresponding x-coordinate of the graph of g(x).
2. Vertical shift:
The term -1 in the function g(x) = (x - 1)^3 - 1 indicates a vertical shift of one unit downward. This means that every y-coordinate of the graph of f(x) needs to be decreased by one to obtain the corresponding y-coordinate of the graph of g(x).
Overall, to obtain the graph of g from the graph of f:
- Shift every point of the graph of f one unit to the right.
- Shift every point of the graph of f one unit downward.
Applying these transformations, you can plot the graph of f(x) = x^3 and then transform each point according to the shifts described above to plot the graph of g(x) = (x - 1)^3 - 1.