Question

complete the table to investigate dilations of exponential functions.

\\(\

$$\begin{array}{|c|c|c|c|}\\hline x & 2^x & 3 \\cdot 2^x & 2^{3x} \\\\\\hline -2 & \\frac{1}{4} & \\frac{3}{4} & \\frac{1}{64} \\\\\\hline -1 & \\frac{1}{2} & \\frac{3}{2} & \\frac{1}{8} \\\\\\hline 0 & a & b & c \\\\\\hline 1 & d & e & f \\\\\\hline 2 & 4 & 12 & 64 \\\\\\hline\\end{array}$$

\\)

which function represents a vertical stretch of an exponential function?
\\(y = 2^x\\)
\\(y = 3 \cdot 2^x\\)
\\(y = 2^{3x}\\)

which function represents a horizontal compression of an exponential function?
\\(y = 2^x\\)
\\(y = 3 \cdot 2^x\\)
\\(y = 2^{3x}\\)

Explanation:

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<reasoning>

Evaluate the table values at x = 0

Using the Exponential Evaluation knowledge point
\[

$$\begin{aligned} a &= 2^0 = 1\\ b &= 3 \cdot 2^0 = 3 \cdot 1 = 3\\ c &= 2^{3(0)} = 2^0 = 1 \end{aligned}$$

\]

Evaluate the table values at x = 1

Using the Exponential Evaluation knowledge point
\[

$$\begin{aligned} d &= 2^1 = 2\\ e &= 3 \cdot 2^1 = 3 \cdot 2 = 6\\ f &= 2^{3(1)} = 2^3 = 8 \end{aligned}$$

\]

Identify the vertical stretch

Using the Exponential Function Transformations knowledge point
A vertical stretch of a parent function \(y = f(x)\) is represented by \(y = a \cdot f(x)\) where \(|a| > 1\).
Comparing \(y = 3 \cdot 2^x\) to the parent function \(y = 2^x\), the factor \(3\) multiplies the output values, representing a vertical stretch by a factor of \(3\).

Identify the horizontal compression

Using the Exponential Function Transformations knowledge point
A horizontal compression of a parent function \(y = f(x)\) is represented by \(y = f(b \cdot x)\) where \(|b| > 1\).
Comparing \(y = 2^{3x}\) to the parent function \(y = 2^x\), the input \(x\) is multiplied by \(3\), representing a horizontal compression by a factor of \(\frac{1}{3}\).
</reasoning>

<answer>

Question 1

• \(a = 1\)
• \(b = 3\)
• \(c = 1\)
• \(d = 2\)
• \(e = 6\)
• \(f = 8\)

Question 2

<mcq-option>(A) \(y = 2^x\)</mcq-option>
<mcq-correct>(B) \(y = 3 \cdot 2^x\)</mcq-correct>
<mcq-option>(C) \(y = 2^{3x}\)</mcq-option>

Question 3

<mcq-option>(A) \(y = 2^x\)</mcq-option>
<mcq-option>(B) \(y = 3 \cdot 2^x\)</mcq-option>
<mcq-correct>(C) \(y = 2^{3x}\)</mcq-correct>
</answer>

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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Function Transformations"
]
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</post_analysis>

Answer:

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<reasoning>

Evaluate the table values at x = 0

Using the Exponential Evaluation knowledge point
\[

$$\begin{aligned} a &= 2^0 = 1\\ b &= 3 \cdot 2^0 = 3 \cdot 1 = 3\\ c &= 2^{3(0)} = 2^0 = 1 \end{aligned}$$

\]

Evaluate the table values at x = 1

Using the Exponential Evaluation knowledge point
\[

$$\begin{aligned} d &= 2^1 = 2\\ e &= 3 \cdot 2^1 = 3 \cdot 2 = 6\\ f &= 2^{3(1)} = 2^3 = 8 \end{aligned}$$

\]

Identify the vertical stretch

Using the Exponential Function Transformations knowledge point
A vertical stretch of a parent function \(y = f(x)\) is represented by \(y = a \cdot f(x)\) where \(|a| > 1\).
Comparing \(y = 3 \cdot 2^x\) to the parent function \(y = 2^x\), the factor \(3\) multiplies the output values, representing a vertical stretch by a factor of \(3\).

Identify the horizontal compression

Using the Exponential Function Transformations knowledge point
A horizontal compression of a parent function \(y = f(x)\) is represented by \(y = f(b \cdot x)\) where \(|b| > 1\).
Comparing \(y = 2^{3x}\) to the parent function \(y = 2^x\), the input \(x\) is multiplied by \(3\), representing a horizontal compression by a factor of \(\frac{1}{3}\).
</reasoning>

<answer>

Question 1

• \(a = 1\)
• \(b = 3\)
• \(c = 1\)
• \(d = 2\)
• \(e = 6\)
• \(f = 8\)

Question 2

<mcq-option>(A) \(y = 2^x\)</mcq-option>
<mcq-correct>(B) \(y = 3 \cdot 2^x\)</mcq-correct>
<mcq-option>(C) \(y = 2^{3x}\)</mcq-option>

Question 3

<mcq-option>(A) \(y = 2^x\)</mcq-option>
<mcq-option>(B) \(y = 3 \cdot 2^x\)</mcq-option>
<mcq-correct>(C) \(y = 2^{3x}\)</mcq-correct>
</answer>

<post_analysis>
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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Function Transformations"
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