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What is the equation of the translated function, [tex]$g(x)$[/tex], if [tex]$f(x)=x^2$[/tex]?

A. [tex]$g(x)=(x-4)^2+6$[/tex]
B. [tex][tex]$g(x)=(x+6)^2-4$[/tex][/tex]
C. [tex]$g(x)=(x-6)^2-4$[/tex]
D. [tex]$g(x)=(x+4)^2+6$[/tex]

Sagot :

GinnyAnswer
To determine the correct translated function [tex]\( g(x) \)[/tex] given the function [tex]\( f(x) = x^2 \)[/tex], we will analyze each provided option. Let's review each option step-by-step to see which transformation matches the desired translation.

1. Option A: [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]

- Horizontal Shift: The term [tex]\( (x-4) \)[/tex] indicates a shift to the right by 4 units.
- Vertical Shift: The term [tex]\( +6 \)[/tex] indicates a shift upward by 6 units.

Thus, this transformation represents a right shift of 4 units and an upward shift of 6 units for the original function [tex]\( f(x) = x^2 \)[/tex].

2. Option B: [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]

- Horizontal Shift: The term [tex]\( (x+6) \)[/tex] indicates a shift to the left by 6 units.
- Vertical Shift: The term [tex]\( -4 \)[/tex] indicates a shift downward by 4 units.

So, this transformation shifts the original function left by 6 units and downward by 4 units.

3. Option C: [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]

- Horizontal Shift: The term [tex]\( (x-6) \)[/tex] indicates a shift to the right by 6 units.
- Vertical Shift: The term [tex]\( -4 \)[/tex] indicates a shift downward by 4 units.

This transformation shifts the original function right by 6 units and downward by 4 units.

4. Option D: [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]

- Horizontal Shift: The term [tex]\( (x+4) \)[/tex] indicates a shift to the left by 4 units.
- Vertical Shift: The term [tex]\( +6 \)[/tex] indicates a shift upward by 6 units.

This transformation shifts the original function left by 4 units and upward by 6 units.

Based on the analysis:

- The translated function [tex]\( g(x) = (x-4)^2 + 6 \)[/tex] clearly matches a right shift of 4 units and an upward shift of 6 units.

Therefore, the correct equation of the translated function [tex]\( g(x) \)[/tex] is:

[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]
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