Answer :
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.
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.
