Question

Write your answer in a(x-h)^2 + k
Find g(x) where g(x) is the translation 4 units left of f(x) =x^2. g(x) = ___
Find g(x) where g(x) is the translation 5 units up of f(x) = x^2. g(x)= __
Find g(x) where g(x) is the reflection across the x-axis of f(x) = x^2. g(x) = ___
Find g(x) where g(x) is the translation 2 units right and 1 unit down of f(x)=x^2. g(x) = ___

Answer 1

To find the function \( g(x) \) after applying the specified translations and transformations to the function \( f(x) = x^2 \), we will use the vertex form of a quadratic function, which is given by \( a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

1. **Translation 4 units left of \( f(x) = x^2 \):**
- Translating 4 units left means we replace \( x \) with \( x + 4 \).
\[
g(x) = (x + 4)^2 = 1(x + 4)^2 + 0
\]
So, \( g(x) = (x + 4)^2 \).

2. **Translation 5 units up of \( f(x) = x^2 \):**
- Translating 5 units up means we add 5 to the entire function.
\[
g(x) = x^2 + 5 = 1(x - 0)^2 + 5
\]
So, \( g(x) = x^2 + 5 \).

3. **Reflection across the x-axis of \( f(x) = x^2 \):**
- Reflecting across the x-axis means we take the negative of the function.
\[
g(x) = -x^2 = -1(x - 0)^2 + 0
\]
So, \( g(x) = -x^2 \).

4. **Translation 2 units right and 1 unit down of \( f(x) = x^2 \):**
- Translating 2 units right means we replace \( x \) with \( x - 2 \) and translating 1 unit down means we subtract 1 from the entire function.
\[
g(x) = (x - 2)^2 - 1 = 1(x - 2)^2 - 1
\]
So, \( g(x) = (x - 2)^2 - 1 \).

To summarize:

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