Answer: The point (0, 0) corresponds to the point (4, -7)
Step-by-step explanation:
We have the function f(x) = x^3
And we transform this to get:
g(x) = (x - 4)^3 - 7
Here we have a vertical translation and a horizontal shift, let's define these two shifts:
Vertical shift.
If we have a function f(x), a vertical shift of N units is written as:
g(x) = f(x) + N
This will move the graph of f(x) up or down a distance of N units.
if N is positive, then the shift is upwards
if N is negative, then the shift is downwards.
Horizontal shift.
If we have a function f(x), a horizontal shift of N units is written as:
g(x) = f(x + N)
This will move the graph of f(x) to the right or left a distance of N units.
if N is positive, then the shift is to the left
if N is negative, then the shift is to the right.
Then if we start with f(x), g(x) is a translation of 7 units down and 4 units to the right.
This means that any point (x, y) that belongs to the graph of f(x), after the transformation will be (x + 4, y - 7)
Then the point (0, 0) of the original function corresponds to the point (0 + 4, 0 - 7) = (4, - 7) of the function g(x)
We can check this, we need to evaluate the function g(x) in x = 4
g(4) = (4 - 4)^3 - 7= 0 - 7 = -7
g(4) = -7
Then the point (4, -7) belongs to the graph of g(x).
