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:

Evaluate the table values at x = 0

Using the Exponential Evaluation knowledge point

$$
LATEXBLOCK0
$$

Evaluate the table values at x = 1

Using the Exponential Evaluation knowledge point

$$
LATEXBLOCK1
$$

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 \(\frac{1}{3}\).

Answer:

Question 1

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

Question 2

• (A) \(y = 2^x\)
• (B) \(y = 3 \cdot 2^x\) (Correct answer)
• (C) \(y = 2^{3x}\)

Question 3

• (A) \(y = 2^x\)
• (B) \(y = 3 \cdot 2^x\)
• (C) \(y = 2^{3x}\) (Correct answer)