Question

enter the correct answer in the box.
function g has the same a value as function f, but its vertex is 2 units below and 3 units to the left.
$f(x) = x^2 - 4x - 32$
write the vertex form of the equation modeling function g.
$g(x) = $

Explanation:

Step1: Find vertex of \( f(x) \)

First, convert \( f(x) = x^2 - 4x - 32 \) to vertex form by completing the square.
\( f(x) = x^2 - 4x + 4 - 4 - 32 \)
\( f(x) = (x - 2)^2 - 36 \)
So, vertex of \( f(x) \) is \( (2, -36) \).

Step2: Determine vertex of \( g(x) \)

Vertex of \( g(x) \) is 3 units left (subtract 3 from \( x \)-coordinate) and 2 units down (subtract 2 from \( y \)-coordinate) from \( f(x) \)’s vertex.
New \( x \)-coordinate: \( 2 - 3 = -1 \)
New \( y \)-coordinate: \( -36 - 2 = -38 \)
Vertex of \( g(x) \) is \( (-1, -38) \).

Step3: Write vertex form of \( g(x) \)

Since \( g(x) \) has the same \( a \)-value (\( a = 1 \)) as \( f(x) \), vertex form is \( g(x) = a(x - h)^2 + k \), where \( (h, k) = (-1, -38) \) and \( a = 1 \).
Substitute: \( g(x) = (x - (-1))^2 + (-38) \)
Simplify: \( g(x) = (x + 1)^2 - 38 \)

Answer:

\( (x + 1)^2 - 38 \)