Question

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The graph shows f(x) and its transformation g(x) .

What is the function equation for g(x)?
g(x)=2^x+3
g(x)=2^x−3
g(x)=2^x−3
g(x)=2^x+3

Answer 1

Answer:

$g(x) = 2^{x-3}$ is the correct function equation that shows the corresponding transformation as shown in figure.

Step-by-step explanation:

Let us consider the function f(x) = bˣ. Lets add a constant c to the input of parent function i.e f(x) = bˣ. It would produce a horizontal translation move by c units, along x axis .

If c > 0, there will be translation to the "left" by "c" units.

If c < 0, there will be translation to the "right" by "c" units.

So, by analyzing the coordinate locations of the function f(x) = 2ˣ and g(x), it is clear that $g(x) = 2^{x-3}$ is the correct function equation that shows the corresponding transformation. Because the g(x) also does has a translation to the 'right'.

It can be further verified as:

Lets check the coordinate locations of f(x) = 2ˣ and $g(x) = 2^{x-3}$ at the the points y = 1, y = 2 and y = 4

At y = 1, the function f(x) = 2ˣ has the location at (0, 1) as 2ˣ = 2° ⇒ x = 0.

At y = 1, the function $g(x) = 2^{x-3}$ has the location at (3, 1) as $2^{0} = 2^{x-3}$ ⇒ x-3 = 0 ⇒ x = 3. It has a translation to the "right".

At y = 2, the function f(x) = 2ˣ has the location at (1, 2) as 2ˣ = 2¹ ⇒ x = 1.

At y = 2, the function $g(x) = 2^{x-3}$ has the location at (4, 2) as $2^{1} = 2^{x-3}$ ⇒ x-3 = 1 ⇒ x = 4. It has a translation to the "right".

At y = 4, the function f(x) = 2ˣ has the location at (2, 4) as 2ˣ = 2² ⇒ x = 2.

At y = 4, the function $g(x) = 2^{x-3}$ has the location at (5, 4) as $2^{2} = 2^{x-3}$ ⇒ x-3 = 2 ⇒ x = 5. It has a translation to the "right".

So, it is clear that only $g(x) = 2^{x-3}$ holds true and have a translation to the "right" by "c" units.

So, $g(x) = 2^{x-3}$ is the correct function equation that shows the corresponding transformation as shown in figure.

Keywords: transformation, function

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