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

To express the transformations of the function \( f(x) = x^2 \) in the form \( g(x) = a(x - h)^2 + k \), we will apply the appropriate transformations according to each question's instructions.

#1

Translation 4 units left of \( f(x) = x^2 \)
When translating a function left by \( h \) units, we replace \( x \) with \( x + h \). So: \[ g(x) = (x + 4)^2 \] In the form \( a(x - h)^2 + k \): \[ g(x) = 1(x - (-4))^2 + 0 \]

#2

Translation 5 units up of \( f(x) = x^2 \)
When translating a function up by \( k \) units, we add \( k \) to the function: \[ g(x) = x^2 + 5 \] In the form \( a(x - h)^2 + k \): \[ g(x) = 1(x - 0)^2 + 5 \]

#3

Reflection across the x-axis of \( f(x) = x^2 \)
To reflect across the x-axis, we multiply the function by -1: \[ g(x) = -x^2 \] In the form \( a(x - h)^2 + k \): \[ g(x) = -1(x - 0)^2 + 0 \]

#4

Translation 2 units right and 1 unit down of \( f(x) = x^2 \)
First, we translate 2 units to the right by replacing \( x \) with \( x - 2 \). Then, we translate 1 unit down by subtracting 1: \[ g(x) = (x - 2)^2 - 1 \] In the form \( a(x - h)^2 + k \): \[ g(x) = 1(x - 2)^2 - 1 \]

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. \( g(x) = 1(x - 2)^2 - 1 \)