Question

The function g(x) = 2^x. The function f(x) = 2^x+k and k < 0. Which of the following statements is true? A) The graph of f(x) is shifted k units to the left of the graph of g(x). B) The graph of f(x) is shifted k units to the right of the graph of g(x). C) The graph of f(x) is shifted k units above the graph of g(x). D) The graph of f(x) is shifted k units below the graph of g(x).

Answer 1

B) The graph of f(x) is shifted k units to the right of the graph of g(x).

Explanation:

We given the functions
`g(x) = 2^x` and
`f(x)=2^(x+k)` where
`k <0`.

We know that horizontal depends on the value of x and are when

`g(x)=f(x+h)` when graph is shifted to left; and

`g(x)=f(x-h)` when graph is shifted to right.

Given the above functions, when k is considered positive then f(x) becomes
`f(x)=2^(x+k)=g(x-k)`.

Therefore, the graph of f(x) is shifted k units to the right of the graph of g(x).

Answer 2

Option D)

Explanation:

If the graph of the function
`f(x)=g(x+h) +b` represents the transformations made to the graph of
`y= g(x)` then, by definition:

If
`b> 0` the graph moves vertically b units up

If
`b <0` the graph moves vertically b units down

If
`h>0` the graph moves horizontally h units to the left

If
`h> 1` the graph moves horizontally h units to the rigth

In this problem we have the function
`f(x)=2^x+k` and our parent function is
`g(x) = 2^x`

therefore it is true that
`b =k<0` and
`h= 0`

Then "The graph of f(x) is shifted k units below the graph of g(x)".

The answer is Option D)

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Note:

If the function are:

`g(x) = 2^x\\\\f(x) = 2^(x+k)`

Then
`k = h` and
`h<0`. This means that the function f(x) shifts k units to the right of the function g(x)

And the answer would be the option B)