Question

The graph of the parent function f(x)=x^3 is translated to from the graph of g(x)=(x+3)^3 - 4 The point (0,0) on the graph of f(x) corresponds to which point on the graph of g(x)?

Answer 1

The point (0,0) on the graph of f(x) corresponds to point (-3, -4) on the graph of g(x).

Explanation:

We have to ask ourselves the question "what are the characteristics of the point (0,0) on the graph of f(x)?"

One way to find the corresponding point on the graph of g(x), is to demand that the derivative of f(x) at (0,0) must be equal to the derivative of g(x) at the corresponding point.

The derivative of f(x) is

`(df(x))/(x) =(dx^3)/(dx) =3x^2`

and at point (0,0) the derivative is

`(df(0))/(x)=3(0)^2=0.`

the derivative of f(x) is zero at point (0,0).

The derivative of g(x) is

`(dg(x))/(x)=(d(x+3)^3-4)/(x)=3(x+3)^2`

Now we need to ask our selves "at what point is the derivative of g(x) equal to 0 (derivative of f(x) at (0,0) )?"

The answer is the equation

`(dg(x))/(x)=3(x+3)^2=0`

We solve for
`x`

`3(x+3)^2=0`

`(x+3)=0`

`\boxed{x=-3}`

The derivative of
`g(x)` is zero at
`x=-3`, and now we just need to find the corresponding y-coordinate:

`y=g(-3)=(-3+3)^3-4`

`\boxed{ y=-4}`

Thus, the corresponding coordinates of the point on g(x) is (-3, -4).