Answer :
Let's carefully compare the given functions \( f(x) = x^2 + 4 \) and \( g(y) = y^2 + 4 \).
1. Understanding the Functions:
- The function \( f(x) = x^2 + 4 \) describes a parabola that opens upwards, with its vertex at \((0, 4)\).
- Similarly, the function \( g(y) = y^2 + 4 \) describes a parabola that opens upwards, with its vertex at \((0, 4)\).
2. Graphing the Functions:
- By graphing \( f(x) \) and \( g(y) \), you will see that both functions produce parabolas that look identical; they both have the same vertical translation up by 4 units.
3. Checking for Reflections:
- For Reflection over the line \( y = 1 \): This would mean that the graphs should be symmetric about the line \( y = 1 \). However, neither of these functions shows such symmetry. They are symmetric about their respective axes.
- For Reflection over the \( y \)-axis: This would involve changing \( x \) to \(-x\) but since \( g(y) \) does not involve \( x \), this does not apply.
- For Reflection over the \( x \)-axis: This would involve changing \( y \) to \(-y\), but since both are parabolas opening upward, there is no such reflection.
- For Reflection over the line \( y = x \): This means swapping \( x \) and \( y \). However, if you swap \( x \) and \( y \) in \( g(y) \), you would end up with \( y^2 + 4 \), which remains the same as \( f(x) \).
4. Conclusion:
- Since \( f(x) \) and \( g(y) \) are identical in form, both represent the same parabola translated up by 4 units and there is no reflection over any axis or line between them.
Thus, the correct description of the difference between the graphs of \( f(x) = x^2 + 4 \) and \( g(y) = y^2 + 4 \) is that there is no reflection. Therefore, the answer is:
None of the given options correctly describe the difference. There is no reflection involved.
1. Understanding the Functions:
- The function \( f(x) = x^2 + 4 \) describes a parabola that opens upwards, with its vertex at \((0, 4)\).
- Similarly, the function \( g(y) = y^2 + 4 \) describes a parabola that opens upwards, with its vertex at \((0, 4)\).
2. Graphing the Functions:
- By graphing \( f(x) \) and \( g(y) \), you will see that both functions produce parabolas that look identical; they both have the same vertical translation up by 4 units.
3. Checking for Reflections:
- For Reflection over the line \( y = 1 \): This would mean that the graphs should be symmetric about the line \( y = 1 \). However, neither of these functions shows such symmetry. They are symmetric about their respective axes.
- For Reflection over the \( y \)-axis: This would involve changing \( x \) to \(-x\) but since \( g(y) \) does not involve \( x \), this does not apply.
- For Reflection over the \( x \)-axis: This would involve changing \( y \) to \(-y\), but since both are parabolas opening upward, there is no such reflection.
- For Reflection over the line \( y = x \): This means swapping \( x \) and \( y \). However, if you swap \( x \) and \( y \) in \( g(y) \), you would end up with \( y^2 + 4 \), which remains the same as \( f(x) \).
4. Conclusion:
- Since \( f(x) \) and \( g(y) \) are identical in form, both represent the same parabola translated up by 4 units and there is no reflection over any axis or line between them.
Thus, the correct description of the difference between the graphs of \( f(x) = x^2 + 4 \) and \( g(y) = y^2 + 4 \) is that there is no reflection. Therefore, the answer is:
None of the given options correctly describe the difference. There is no reflection involved.
