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The function [tex]$y=(x+4)^4[tex]$[/tex] is a transformation of the graph of the parent function [tex]$[/tex]y=x^4$[/tex]. How is the zero of the parent function affected by the transformation?

A. The zero of the function is shifted 4 units to the right.
B. The zero of the function is shifted 4 units down.
C. The zero of the function is shifted 4 units to the left.
D. The zero of the function is shifted 4 units up.


Sagot :

To analyze how the zero of the parent function \( y = x^4 \) is affected by the transformation to \( y = (x + 4)^4 \), let's first identify the zero of the parent function and understand the transformation applied.

1. Identifying the Zero of the Parent Function:
The parent function is \( y = x^4 \). To find the zero of this function, we set \( y = 0 \):
[tex]\[ x^4 = 0 \][/tex]
Solving for \( x \), we get:
[tex]\[ x = 0 \][/tex]
Hence, the zero of the function \( y = x^4 \) is at \( x = 0 \).

2. Understanding the Transformation:
The given transformed function is \( y = (x + 4)^4 \). This transformation involves \( (x + 4) \). When we add a constant inside the function, it results in a horizontal shift. Specifically, adding a positive constant \( +4 \) inside the parentheses shifts the graph to the left by that number of units.

3. Determining the New Zero:
To find the new zero, we set the transformed function to zero and solve for \( x \):
[tex]\[ (x + 4)^4 = 0 \][/tex]
Solving for \( x \), we get:
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4 \][/tex]
Therefore, the zero of the transformed function \( y = (x + 4)^4 \) is at \( x = -4 \).

4. Comparing the Zero of the Parent and Transformed Functions:
- The original zero was at \( x = 0 \).
- The new zero is at \( x = -4 \), which means the zero has been shifted 4 units to the left from the original position.

Therefore, the correct answer is:

C. The zero of the function is shifted 4 units to the left.