Question

Write your answer in the form a(x−h)2+k
given h is zero and a is an invisible 1.

#1 Find g(x)
where g(x)
is the translation vertically 4 units 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 up and then reflected across the x-axis f(x)=x2
g(x) =

Answer 1

To express the functions in the form \( a(x - h)^2 + k \), where \( h = 0 \) and \( a = 1 \), we have the general form \( (x - 0)^2 + k \) or simply \( x^2 + k \). Below are the required transformations based on the given functions:

#1 Find \( g(x) \) where \( g(x) \) is the translation vertically 4 units of \( f(x) = x^2 \).

To translate \( f(x) = x^2 \) vertically 4 units up, we add 4:
\[ g(x) = x^2 + 4 \]

#2 Find \( g(x) \) where \( g(x) \) is the translation 5 units up of \( f(x) = x^2 \).

Similarly, translating \( f(x) = x^2 \) vertically 5 units up:
\[ g(x) = x^2 + 5 \]

#3 Find \( g(x) \) where \( g(x) \) is the reflection across the x-axis of \( f(x) = x^2 \).

Reflecting across the x-axis means multiplying the function by -1:
\[ g(x) = -x^2 \]

In the required form:
\[ g(x) = 1(x - 0)^2 - 0 \]

#4 Find \( g(x) \) where \( g(x) \) is the translation 2 units up and then reflected across the x-axis of \( f(x) = x^2 \).

First, translate vertically 2 units up:
\[ f(x) = x^2 \Rightarrow g(x) = x^2 + 2 \]

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

In the required form:
\[ g(x) = 1(x - 0)^2 - 2 \]

To summarize the answers:
1. \( g(x) = x^2 + 4 \)
2. \( g(x) = x^2 + 5 \)
3. \( g(x) = -x^2 \)
4. \( g(x) = -x^2 - 2 \)