Answer :
Let's start with the given function [tex]\( f(x) = x^2 \)[/tex].
### Step 1: Reflecting [tex]\( f(x) \)[/tex] Across the [tex]\( x \)[/tex]-Axis
When we reflect a function across the [tex]\( x \)[/tex]-axis, we multiply the entire function by [tex]\(-1\)[/tex]. Thus, reflecting [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis gives us the new function:
[tex]\[ g(x) = -f(x) = -x^2 \][/tex]
### Step 2: Translating [tex]\( g(x) \)[/tex] 3 Units Upward
The second step is to translate this reflected function [tex]\( g(x) \)[/tex] 3 units in the positive [tex]\( y \)[/tex]-direction (upward). Translating a function upward involves adding a constant to the function. Therefore, we add 3 to [tex]\( g(x) \)[/tex]:
[tex]\[ h(x) = g(x) + 3 = -x^2 + 3 \][/tex]
### Conclusion
After reflecting the function across the [tex]\( x \)[/tex]-axis and then translating it 3 units upward, we obtain the new function:
[tex]\[ h(x) = -x^2 + 3 \][/tex]
Hence, the equation of [tex]\( h(x) \)[/tex] is:
[tex]\[ \boxed{-x^2 + 3} \][/tex]
### Step 1: Reflecting [tex]\( f(x) \)[/tex] Across the [tex]\( x \)[/tex]-Axis
When we reflect a function across the [tex]\( x \)[/tex]-axis, we multiply the entire function by [tex]\(-1\)[/tex]. Thus, reflecting [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis gives us the new function:
[tex]\[ g(x) = -f(x) = -x^2 \][/tex]
### Step 2: Translating [tex]\( g(x) \)[/tex] 3 Units Upward
The second step is to translate this reflected function [tex]\( g(x) \)[/tex] 3 units in the positive [tex]\( y \)[/tex]-direction (upward). Translating a function upward involves adding a constant to the function. Therefore, we add 3 to [tex]\( g(x) \)[/tex]:
[tex]\[ h(x) = g(x) + 3 = -x^2 + 3 \][/tex]
### Conclusion
After reflecting the function across the [tex]\( x \)[/tex]-axis and then translating it 3 units upward, we obtain the new function:
[tex]\[ h(x) = -x^2 + 3 \][/tex]
Hence, the equation of [tex]\( h(x) \)[/tex] is:
[tex]\[ \boxed{-x^2 + 3} \][/tex]
