Write your answer in the form a(x−h)2+k

To express the function \( f(x) = x^2 \) in the form \( a(x - h)^2 + k \), we can use the format for transformations based on the vertex form of a quadratic function.

1. Translation 4 units left of \( f(x) = x^2 \):

   • To translate left, we replace \( x \) with \( x + 4 \).
   • Therefore, \( g(x) = (x + 4)^2 \).
   • In vertex form: \[ g(x) = 1(x - (-4))^2 + 0 \quad \Rightarrow \quad g(x) = 1(x + 4)^2 + 0 \]

   So, \[ g(x) = 1(x + 4)^2 + 0 \]

2. Translation 5 units up of \( f(x) = x^2 \):

   • To translate up, we add 5 to the function.
   • Therefore, \( g(x) = x^2 + 5 \).
   • In vertex form: \[ g(x) = 1(x - 0)^2 + 5 \]

   So, \[ g(x) = 1(x - 0)^2 + 5 \]

3. Reflection across the x-axis of \( f(x) = x^2 \):

   • To reflect across the x-axis, we multiply the function by \(-1\).
   • Therefore, \( g(x) = -x^2 \).
   • In vertex form: \[ g(x) = -1(x - 0)^2 + 0 \]

   So, \[ g(x) = -1(x - 0)^2 + 0 \]

4. Translation 2 units right and 1 unit down of \( f(x) = x^2 \):

   • To translate right, we replace \( x \) with \( x - 2 \).
   • Then, to translate down, we subtract 1 from the function.
   • Therefore, \( g(x) = (x - 2)^2 - 1 \).
   • In vertex form: \[ g(x) = 1(x - 2)^2 - 1 \]

   So, \[ g(x) = 1(x - 2)^2 - 1 \]

In summary:

1. \( g(x) = 1(x + 4)^2 + 0 \)
2. \( g(x) = 1(x - 0)^2 + 5 \)
3. \( g(x) = -1(x - 0)^2 + 0 \)
4. \( g(x) = 1(x - 2)^2 - 1 \)