Question

Let f(x)=4^x and g(x)=2(4)^x−1 .

Which transformations are needed to transform the graph of f(x) to the graph of g(x) ?

Select each correct answer.

A.horizontal translation 1 unit left

B.vertical translation 1 unit up

C.horizontal translation 1 unit right

D.vertical compression by a factor of 1/2

E.vertical stretch by a factor of 2

F.vertical translation 2 units up

Answer 1

Attached is a very useful chart for translations.

A number added or subtracted outside of parentheses shifts the graph vertically up (if number is added) or down (if number is subtracted).
As you can see in the chart,
f(x) + k shifts the graph up k units, and
f(x) - k shifts the graph down k units.

A number that is multiplied to x outside parentheses stretches or shrinks the graph vertically. If that number is greater than 1, then it stretches the graph. If the number is a fraction or between 0 and 1, then it shrinks the graph vertically.

As you can see in the chart,
a * f(x), where a>1 stretches the graph f(x) vertically by a factor a.
a * f(x), where a is between 0 and 1 shrinks the graph f(x) vertically by a factor of a.

Looking at your problem, to get f(x)=4^x to g(x)=2(4)^x−1, a number 2 is multiplied outside the function f(x), which results in a vertical stretch by a factor of 2.
A number 1 is also subtracted from the function, which vertically translates the graph 1 unit down.