Question 9 of 10 How would you describe the difference between the graphs of \( f(x)=x^{2}+4 \) and \( g(y)=y^{2}+4 \) ? A. \( g(y) \) is a reflection of \( f(x) \) over the \( y \)-axis. B. \( g(y) \) is a reflection of \( f(x) \) over the line \( y=1 \) D. \( g(y) \) is a reflection of \( f(x) \) over the line \( y=x \). Deflection of \( f(x) \) over the \( x \)-axis.

To analyze the functions \( f(x) = x^2 + 4 \) and \( g(y) = y^2 + 4 \), we need to understand their graphs and how they relate to each other. 1. **Graph of \( f(x) = x^2 + 4 \)**: - This is a parabola that opens upwards with its vertex at the point \( (0, 4) \). The graph is symmetric about the \( y \)-axis. 2. **Graph of \( g(y) = y^2 + 4 \)**: - This is also a parabola that opens upwards, but it is expressed in terms of \( y \). If we rewrite it in terms of \( x \), we can express it as \( x = y^2 + 4 \). This means that for each value of \( y \), there is a corresponding value of \( x \). Now, let's analyze the options: A. **\( g(y) \) is a reflection of \( f(x) \) over the \( y \)-axis.** - This is incorrect. The graphs are not reflections over the \( y \)-axis; they are different representations of parabolas. B. **\( g(y) \) is a reflection of \( f(x) \) over the line \( y=1 \).** - This is also incorrect. The graphs do not reflect over the line \( y=1 \). C. **\( g(y) \) is a reflection of \( f(x) \) over the line \( y=x \).** - This is correct. The graph of \( g(y) \) can be obtained by reflecting the graph of \( f(x) \) over the line \( y=x \). D. **Deflection of \( f(x) \) over the \( x \)-axis.** - This is incorrect. The graphs do not reflect over the \( x \)-axis. Thus, the correct answer is: **C. \( g(y) \) is a reflection of \( f(x) \) over the line \( y=x \).**