Question

use the graph f(x)=log x to graph g(x)=log(x+2)

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = f(x + h) \), if \( h>0 \), the graph of \( f(x) \) shifts left by \( h \) units; if \( h < 0 \), it shifts right by \(|h|\) units. Here, \( g(x)=\log(x + 2)=f(x+2) \) where \( f(x)=\log x \) and \( h = 2>0 \).

Step2: Analyze Key Points of \( f(x)=\log x \)

• Vertical asymptote: \( x = 0 \) (y - axis).
• Passes through \( (1,0) \) since \( \log(1)=0 \) (assuming base 10 or natural log, the property holds).
• Passes through \( (10,1) \) for base 10: \( \log(10)=1 \), or \( (e,1) \) for natural log \( \ln(e)=1 \).

Step3: Apply Horizontal Shift to Key Points

• Vertical asymptote of \( f(x) \) is \( x = 0 \). Shifting left by 2 units, vertical asymptote of \( g(x) \) becomes \( x=0 - 2=- 2 \).
• Point \( (1,0) \) on \( f(x) \): shift left by 2 units, new point is \( (1 - 2,0)=(-1,0) \).
• Point \( (10,1) \) (for base 10) on \( f(x) \): shift left by 2 units, new point is \( (10 - 2,1)=(8,1) \). For natural log, \( (e,1) \) shifts to \( (e - 2,1) \).
• The shape of the logarithmic graph (increasing, concave down) remains the same, just shifted left by 2 units.

Answer:

To graph \( g(x)=\log(x + 2) \) from \( f(x)=\log x \), shift the graph of \( f(x) \) 2 units to the left. The vertical asymptote moves from \( x = 0 \) to \( x=-2 \), and key points like \( (1,0) \) on \( f(x) \) move to \( (-1,0) \), \( (10,1) \) (base 10) move to \( (8,1) \) (or corresponding natural log points shift similarly). The general shape (increasing, concave - down for \( \log x \) with base \(>1\)) is preserved.