Question

question 1-4
how do the graphs of ( f(x) = 2x + 1 ) and ( g(x) = 2(x + 3) + 1 ) compare?

• the graph of ( g ) is the graph of ( f ) translated 3 units left.
• the graph of ( g ) is the graph of ( f ) translated 3 units up.
• the graph of ( g ) is the graph of ( f ) translated 3 units right.
• the graph of ( g ) is the graph of ( f ) translated 3 units down.

Explanation:

Step1: Recall translation rules

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

Step2: Analyze \( f(x) \) and \( g(x) \)

Given \( f(x)=2x + 1 \) and \( g(x)=2(x + 3)+1 \). Rewrite \( g(x) \) as \( g(x)=f(x + 3) \) (since \( f(x)=2x + 1 \), so \( f(x + 3)=2(x + 3)+1 \)). Here, \( h = 3>0 \), so by horizontal translation rule, \( g(x) \) is \( f(x) \) shifted left 3 units.

Answer:

The graph of g is the graph of f translated 3 units left.