Question

use the graph of y = f(x) to graph the function g(x) = f(x + 1) + 6. choose the correct graph of g below. options: a, b, c, d (with respective graphs shown).

Explanation:

Step1: Analyze Horizontal Shift

For \( g(x) = f(x + 1) \), the transformation is a horizontal shift. The rule for \( f(x + h) \) is a shift left by \( h \) units (when \( h>0 \)). Here, \( h = 1 \), so the graph of \( f(x) \) shifts left 1 unit.

Step2: Analyze Vertical Shift

For \( g(x)=f(x + 1)+6 \), after the horizontal shift, we apply a vertical shift. The rule for \( f(x)+k \) is a shift up by \( k \) units (when \( k>0 \)). Here, \( k = 6 \), so the graph after the horizontal shift shifts up 6 units.

Step3: Compare with Options

• Original \( f(x) \) has key points. After shifting left 1 and up 6, the new graph should be higher and to the left. Option A shows a graph shifted left and up (higher y - values), matching the transformations. Options B (lower), C (lower and wrong direction), D (wrong shift direction) do not match.

Answer:

A. The graph corresponding to Option A