To find \( g(x) \) for each of the transformations of \( f(x) = x^2 \) in the form \( a(x-h)^2 + k \), we will apply the corresponding transformations one by one.
#1. Translation 4 units left of \( f(x) = x^2 \):
Translating a graph left involves replacing \( x \) with \( x + h \). In this case, \( h = 4 \): \[ g(x) = (x + 4)^2 \]
#2. Translation 5 units up of \( f(x) = x^2 \):
Translating a graph upwards involves adding \( k \) to the function. Here, \( k = 5 \): \[ g(x) = x^2 + 5 \]
#3. Reflection across the x-axis of \( f(x) = x^2 \):
Reflecting across the x-axis involves multiplying the function by -1: \[ g(x) = -x^2 \]
#4. Translation 2 units right and 1 unit down of \( f(x) = x^2 \):
For a translation right, we replace \( x \) with \( x - h \), where \( h = 2 \). Downwards translation involves subtracting \( k \) from the function, with \( k = 1 \): \[ g(x) = (x - 2)^2 - 1 \]
Putting everything together:
- \( g(x) = (x + 4)^2 \)
- \( g(x) = x^2 + 5 \)
- \( g(x) = -x^2 \)
- \( g(x) = (x - 2)^2 - 1 \)