The translation of the parent function is:
g(x) = ∛(x + 6) + 1
How do translations work?
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)
- If N is positive, the translation is to the left.
- If N is negative, the translation is to the right.
Vertical translation:
For a function f(x), a vertical translation of N units is written as:
g(x) = f(x) + N
- If N is positive the translation is upwards.
- If N is negative the translation is downwards.
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:
https://brainly.com/question/17586310
