The graph of the cube root parent function y = RootIndex 3 StartRoot x EndRoot is translated to form f(x) shown on the graph. On a coordinate plane, a cube root function goes through (negative 7, 0), has an inflection point at (negative 6, 1), and goes through (2, 3).
The graph of the cube root parent function y = RootIndex 3 StartRoot x EndRoot is translated to form f(x) shown on the graph.

On a coordinate plane, a cube root function goes through (negative 7, 0), has an inflection point at (negative 6, 1), and goes through (2, 3).

Which equation represents f(x)?

f(x) = RootIndex 3 StartRoot x + 6 EndRoot + 1
f(x) = RootIndex 3 StartRoot x minus 6 EndRoot + 1
f(x) = RootIndex 3 StartRoot x + 6 EndRoot minus 1
f(x) = RootIndex 3 StartRoot x minus 6 EndRoot minus 1

The translation of the parent function is:

g(x) = ∛(x + 6) + 1

There are two types of translations, these are:

Horizontal translation:

For a function f(x), a horizontal translation of N units is written as:

g(x) = f(x + N)

Vertical translation:

For a function f(x), a vertical translation of N units is written as:

g(x) = f(x) + N

Here we start with the parent function:

f(x) = ∛x

And we apply two translations such that:

g(x) = ∛(x + a) + b

Such that:

g(-7) = 0

g(2) = 3

Then we have the system of equations:

∛(-7 + a) + b = 0

∛(2 + a) + b = 3

From the options, the only ones that make the first option true are:

g(x) = ∛(x + 6) + 1

Such that when evaluated in x = -7 we get:

g(-7) =  ∛(-7 + 6) + 1 =  ∛(-1) + 1 = 0

When evaluated in x = 2 it gives:

∛(2 + 6) + 1 =  ∛(8) + 1 = 2+ 1 = 3

So the only option that meets these conditions is:

g(x) = ∛(x + 6) + 1

Which is a translation to the left of 6 units and up 1 unit.

If you want to learn more about translations, you can read:

link

step-by-step explanation:

the true ones are a b and d