Question

which equation represents the function \\(f(x) = (1.6)^x\\) after it has been translated 5 units up and 9 units to the right?

\\(g(x) = (1.6)^{x + 5} - 9\\)
\\(g(x) = (1.6)^{x + 5} + 9\\)
\\(g(x) = (1.6)^{x - 9} + 5\\)
\\(g(x) = (1.6)^{x + 9} + 5\\)

Explanation:

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<reasoning>

Apply horizontal translation

Using the Function Transformations and Exponential Functions knowledge points
\[

$$\begin{aligned} &f(x) = (1.6)^x\\ &h(x) = f(x - 9) = (1.6)^{x - 9} \end{aligned}$$

\]

Apply vertical translation

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &g(x) = h(x) + 5\\ &g(x) = (1.6)^{x - 9} + 5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(g(x) = (1.6)^{x + 5} - 9\)</mcq-option>
<mcq-option>(B) \(g(x) = (1.6)^{x + 5} + 9\)</mcq-option>
<mcq-correct>(C) \(g(x) = (1.6)^{x - 9} + 5\)</mcq-correct>
<mcq-option>(D) \(g(x) = (1.6)^{x + 9} + 5\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Apply horizontal translation

Using the Function Transformations and Exponential Functions knowledge points
\[

$$\begin{aligned} &f(x) = (1.6)^x\\ &h(x) = f(x - 9) = (1.6)^{x - 9} \end{aligned}$$

\]

Apply vertical translation

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &g(x) = h(x) + 5\\ &g(x) = (1.6)^{x - 9} + 5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(g(x) = (1.6)^{x + 5} - 9\)</mcq-option>
<mcq-option>(B) \(g(x) = (1.6)^{x + 5} + 9\)</mcq-option>
<mcq-correct>(C) \(g(x) = (1.6)^{x - 9} + 5\)</mcq-correct>
<mcq-option>(D) \(g(x) = (1.6)^{x + 9} + 5\)</mcq-option>
</answer>

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