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What value represents the vertical translation from the graph of the parent function [tex]f(x) = x^2[/tex] to the graph of the function [tex]g(x) = (x+5)^2 + 3[/tex]?

A. [tex]\(-5\)[/tex]
B. [tex]\(-3\)[/tex]
C. 3
D. 5

Sagot :

GinnyAnswer
To determine the vertical translation from the graph of the parent function [tex]\( f(x) = x^2 \)[/tex] to the graph of the function [tex]\( g(x) = (x+5)^2 + 3 \)[/tex], follow these steps:

1. Understand the Parent Function:
The parent function in this case is [tex]\( f(x) = x^2 \)[/tex].

2. Analyze the New Function:
The given function is [tex]\( g(x) = (x+5)^2 + 3 \)[/tex].

3. Identify the Components of the Function:
- [tex]\( (x+5)^2 \)[/tex] represents a horizontal shift.
- The constant [tex]\( +3 \)[/tex] at the end represents a vertical shift.

4. Determine the Horizontal Shift:
- The term [tex]\( (x+5) \)[/tex] indicates a horizontal shift 5 units to the left, but this shift does not affect the vertical translation we are interested in.

5. Determine the Vertical Shift:
- The addition of [tex]\( +3 \)[/tex] at the end of the function means the graph is shifted 3 units up vertically.

Therefore, the value that represents the vertical translation from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( g(x) = (x+5)^2 + 3 \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
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