Question

the graphs of (f(x) = 10^x) and its translation, (g(x)), are shown.
what is the equation of (g(x))?

(g(x) = 10^{x-3})
(g(x) = 10^{x+3})
(g(x) = 10^x - 3)
(g(x) = 10^x + 3)

Explanation:

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<reasoning>

Identify corresponding points on both graphs

Compare the labeled points on the graph of \(f(x) = 10^x\) and its translation \(g(x)\):

• \((0, 1) \to (3, 1)\)
• \((1, 10) \to (4, 10)\)
• \((2, 100) \to (5, 100)\)

Determine the horizontal shift

Calculate the change in the \(x\)-coordinates for each corresponding pair:
\[
\Delta x = 3 - 0 = 3
\]
\[
\Delta x = 4 - 1 = 3
\]
\[
\Delta x = 5 - 2 = 3
\]
The graph is translated horizontally to the right by \(3\) units.

Write the equation for the translated function

A horizontal shift of a function \(f(x)\) to the right by \(h\) units is represented by \(f(x - h)\):
\[
g(x) = f(x - 3) = 10^{x - 3}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(g(x) = 10^{x - 3}\)</mcq-correct>
<mcq-option>(B) \(g(x) = 10^{x + 3}\)</mcq-option>
<mcq-option>(C) \(g(x) = 10^x - 3\)</mcq-option>
<mcq-option>(D) \(g(x) = 10^x + 3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify corresponding points on both graphs

Compare the labeled points on the graph of \(f(x) = 10^x\) and its translation \(g(x)\):

• \((0, 1) \to (3, 1)\)
• \((1, 10) \to (4, 10)\)
• \((2, 100) \to (5, 100)\)

Determine the horizontal shift

Calculate the change in the \(x\)-coordinates for each corresponding pair:
\[
\Delta x = 3 - 0 = 3
\]
\[
\Delta x = 4 - 1 = 3
\]
\[
\Delta x = 5 - 2 = 3
\]
The graph is translated horizontally to the right by \(3\) units.

Write the equation for the translated function

A horizontal shift of a function \(f(x)\) to the right by \(h\) units is represented by \(f(x - h)\):
\[
g(x) = f(x - 3) = 10^{x - 3}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(g(x) = 10^{x - 3}\)</mcq-correct>
<mcq-option>(B) \(g(x) = 10^{x + 3}\)</mcq-option>
<mcq-option>(C) \(g(x) = 10^x - 3\)</mcq-option>
<mcq-option>(D) \(g(x) = 10^x + 3\)</mcq-option>
</answer>

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