Question

1. in the function $y = 2^{3x}$, what effect does the 3 in the exponent have?

a. horizontal compression by $\frac{1}{3}$

b. horizontal shift left 3 units

c. vertical compression by $\frac{1}{3}$

d. vertical stretch by 3

Explanation:

Step1: Recall Transformations of Exponential Functions

For a function \( y = a^{bx} \), the coefficient \( b \) inside the exponent affects the horizontal compression/stretch. The general rule is: if we have a parent function \( y = a^x \), and we transform it to \( y = a^{bx} \), the horizontal compression factor is \( \frac{1}{|b|} \) (when \( |b|>1 \), it's a compression; when \( 0<|b|<1 \), it's a stretch).

Step2: Analyze the Given Function

In the function \( y = 2^{3x} \), the parent function is \( y = 2^x \), and here \( b = 3 \). Using the rule from Step 1, the horizontal compression factor is \( \frac{1}{|3|}=\frac{1}{3} \). So the graph of \( y = 2^{3x} \) is a horizontal compression of the graph of \( y = 2^x \) by a factor of \( \frac{1}{3} \).

Let's also eliminate the other options:

• Option b: A horizontal shift left 3 units would be of the form \( y = 2^{x + 3} \), not \( y = 2^{3x} \), so b is incorrect.
• Option c: Vertical compression by \( \frac{1}{3} \) would be of the form \( y=\frac{1}{3}\cdot2^{x} \), not involving the exponent's coefficient in that way, so c is incorrect.
• Option d: Vertical stretch by 3 would be of the form \( y = 3\cdot2^{x} \), not \( y = 2^{3x} \), so d is incorrect.

Answer:

a. Horizontal compression by \( \frac{1}{3} \)