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 ) vertically shifted down 3 units.
b. the graph of ( h ) is the graph of ( g ) horizontally shifted right 3 units.
c. the graph of ( h ) is the graph of ( g ) vertically shifted up 3 units.
d. the graph of ( h ) is the graph of ( g ) horizontally shifted left 3 units.

Explanation:

Step1: Recall Function Transformation Rules

For a function \( y = f(x) \), the transformation \( y = f(x - a) \) (where \( a>0 \)) represents a horizontal shift of the graph of \( f(x) \). The direction of the shift is determined by the value of \( a \): if we have \( f(x - a) \), the graph shifts \( a \) units to the right; if we have \( f(x + a) \) (which can be written as \( f(x - (-a)) \)), the graph shifts \( a \) units to the left. Vertical shifts are of the form \( y = f(x) + k \) (shift up \( k \) units if \( k>0 \), down \( k \) units if \( k<0 \)).

Step2: Analyze the Given Functions

We are given \( g(x)=x^{2} \) and \( h(x)=g(x - 3) \). Substituting \( g(x) \) into the definition of \( h(x) \), we get \( h(x)=(x - 3)^{2} \).

Comparing with the transformation rule for horizontal shifts: for the function \( g(x) \), when we replace \( x \) with \( x - 3 \) to get \( h(x) \), this is a horizontal shift. Since the transformation is of the form \( f(x - a) \) with \( a = 3 \) (positive), the graph of \( h(x) \) is the graph of \( g(x) \) shifted horizontally to the right by 3 units.

Let's analyze the other options to confirm:

• Option A: Vertical shift down 3 units would be \( h(x)=g(x)-3=x^{2}-3 \), which is not our case.
• Option B: Horizontal shift right 3 units matches our analysis of \( h(x)=(x - 3)^{2} \).
• Option C: Vertical shift up 3 units would be \( h(x)=g(x)+3=x^{2}+3 \), not our case.
• Option D: Horizontal shift left 3 units would be \( h(x)=g(x + 3)=(x + 3)^{2} \), which is not our case.

Answer:

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