Question

select the correct answer.
the function ( g(x) = x^2 ) is transformed to obtain function ( h ):
( h(x) = g(x - 3) ).
which statement describes how the graph of ( h ) is different from the graph of ( g )?
a. the graph of ( h ) is the graph of ( g ) horizontally shifted right 3 units.
b. the graph of ( h ) is the graph of ( g ) horizontally shifted left 3 units.
c. the graph of ( h ) is the graph of ( g ) vertically shifted down 3 units.
d. the graph of ( h ) is the graph of ( g ) vertically shifted up 3 units.

Explanation:

Step1: Recall Function Shift Rules

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

Step2: Analyze \( h(x) = g(x - 3) \)

Here, \( g(x)=x^2 \), so \( h(x)=(x - 3)^2 \). Comparing to the horizontal shift rule, \( a = 3>0 \), so the graph of \( h \) is the graph of \( g \) shifted right 3 units horizontally.

Answer:

A. The graph of \( h \) is the graph of \( g \) horizontally shifted right 3 units.