Question

(1 point)
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) =

Answer 1

To find the translations and transformations of the function \( f(x) = x^2 \), we can apply the following rules:

1. **Translation**:
- A translation \( h \) units to the left is achieved by replacing \( x \) with \( x + h \).
- A translation \( h \) units to the right is achieved by replacing \( x \) with \( x - h \).
- A translation \( k \) units up is achieved by adding \( k \) to the function.
- A translation \( k \) units down is achieved by subtracting \( k \) from the function.

2. **Reflection**:
- A reflection across the x-axis is achieved by multiplying the entire function by -1.

Now, let's solve each of the problems one by one.

### 1. Translation 4 Units Left of \( f(x) = x^2 \)

To translate 4 units left:
\[
g(x) = (x + 4)^2
\]

### 2. Translation 5 Units Up of \( f(x) = x^2 \)

To translate 5 units up:
\[
g(x) = x^2 + 5
\]

### 3. Reflection Across the x-axis of \( f(x) = x^2 \)

To reflect across the x-axis:
\[
g(x) = -x^2
\]

### 4. Translation 2 Units Right and 1 Unit Down of \( f(x) = x^2 \)

For a 2 units right translation and then 1 unit down:
\[
g(x) = (x - 2)^2 - 1
\]

### Summary of the Answers

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