Question

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

#1 Find g(x)
where g(x)
is the translation 4 units left of f(x)=x2
g(x) =
#2 Find g(x)
where g(x)
is the translation 5 units up of f(x)=x2
g(x) =
#3 Find g(x)
where g(x)
is the reflection across the x-axis of f(x)=x2
g(x) =
#4 Find g(x)
where g(x)
is the translation 2 units right and 1 unit down of f(x)=x2
g(x) =
a

Answer 1

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