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Which statement is true about the effects of the transformations on the graph of function [tex]\( f \)[/tex] to obtain the graph of function [tex]\( g \)[/tex]? [tex]\( g(x) = f(x-3) + 4 \)[/tex]

A. The graph of function [tex]\( f \)[/tex] is shifted right 3 units and up 4 units.
B. The graph of function [tex]\( f \)[/tex] is shifted right 3 units and down 4 units.
C. The graph of function [tex]\( f \)[/tex] is shifted left 3 units and down 4 units.
D. The graph of function [tex]\( f \)[/tex] is shifted left 3 units and up 4 units.

Sagot :

GinnyAnswer
Let's analyze the given function transformation step-by-step to determine the correct statement about the effects on the graph of function [tex]\( f \)[/tex] to obtain the graph of [tex]\( g \)[/tex].

The given transformation is [tex]\( g(x) = f(x - 3) + 4 \)[/tex].

### Step 1: Horizontal Shift
The term [tex]\( x - 3 \)[/tex] inside the function [tex]\( f \)[/tex]:

1. The expression [tex]\( f(x - h) \)[/tex] indicates a horizontal shift of the graph of [tex]\( f(x) \)[/tex].
2. If [tex]\( h \)[/tex] is positive (i.e., [tex]\( f(x - 3) \)[/tex]), it means the graph of [tex]\( f \)[/tex] is shifted to the right by [tex]\( h \)[/tex] units.
3. Thus, [tex]\( x - 3 \)[/tex] represents a horizontal shift to the right by 3 units.

### Step 2: Vertical Shift
The term [tex]\( + 4 \)[/tex] outside the function [tex]\( f \)[/tex]:

1. The expression [tex]\( f(x) + k \)[/tex] indicates a vertical shift of the graph of [tex]\( f(x) \)[/tex].
2. If [tex]\( k \)[/tex] is positive (i.e., [tex]\( +4 \)[/tex]), it means the graph of [tex]\( f \)[/tex] is shifted upwards by [tex]\( k \)[/tex] units.
3. Thus, [tex]\( +4 \)[/tex] represents a vertical shift upwards by 4 units.

### Conclusion
Combining both transformations:

- The graph of the function [tex]\( f \)[/tex] is shifted to the right by 3 units.
- The graph of the function [tex]\( f \)[/tex] is shifted up by 4 units.

Therefore, the correct statement describing the transformation is:

A. The graph of function [tex]\( f \)[/tex] is shifted right 3 units and up 4 units.

This is the true statement based on the given transformations.
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