Question

1. which transformation applies a horizontal shift to the left?

a. $y = 2^x - 2$
b. $y = 3^{x + 2}$
c. $y = 3^{x - 2}$
d. $y = 2^x + 3$

Explanation:

Step1: Recall horizontal shift rules

For a function \( y = f(x) \), a horizontal shift is given by \( y = f(x + h) \). If \( h>0 \), it's a shift to the left by \( h \) units; if \( h<0 \), it's a shift to the right by \( |h| \) units. Vertical shifts are \( y = f(x)+k \) (up/down).

Step2: Analyze each option

• Option a: \( y = 2^x - 2 \) is a vertical shift down by 2 units (since it's \( f(x)-2 \), \( f(x)=2^x \)).
• Option b: \( y = 3^{x + 2} \). Comparing to \( y = 3^x \) (let \( f(x)=3^x \)), this is \( f(x + 2) \), so \( h = 2>0 \), horizontal shift left by 2 units.
• Option c: \( y = 3^{x - 2} \) is \( f(x - 2) \), \( h=-2 \), horizontal shift right by 2 units.
• Option d: \( y = 2^x + 3 \) is a vertical shift up by 3 units ( \( f(x)+3 \) ).

Answer:

b. \( y = 3^{x+2} \)