Question

what is the equation of the translated function g(x) if f(x) = x²?
○ g(x) = (x + 5)² + 2
○ g(x) = (x + 2)² + 5
○ g(x) = (x - 2)² + 5
○ g(x) = (x - 5)² + 2

Explanation:

Step1: Recall Vertex Form of Parabola

The vertex form of a parabola is \( g(x) = (x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the parent function \( f(x)=x^2 \), the vertex is \((0, 0)\).

Step2: Determine the Vertex of \( g(x) \)

From the graph (and the options), we need to find the vertex of \( g(x) \). Looking at the options, we analyze the horizontal and vertical shifts. A horizontal shift of \( h \) units (right if \( h>0 \), left if \( h<0 \)) and vertical shift of \( k \) units (up if \( k>0 \), down if \( k<0 \)).
Looking at the graph (even though partially visible, from the options), let's check the vertex. For \( g(x)=(x - 2)^2+5 \), the vertex is \((2, 5)\) (since \( h = 2 \), \( k = 5 \)). Let's verify the shifts: from \( f(x)=x^2 \) (vertex \((0,0)\)), moving 2 units to the right (so \( h = 2 \), hence \( (x - 2) \)) and 5 units up (so \( +5 \)).
Let's check other options:

• For \( g(x)=(x + 5)^2+2 \), vertex is \((-5, 2)\) (left 5, up 2) – doesn't match.
• For \( g(x)=(x + 2)^2+5 \), vertex is \((-2, 5)\) (left 2, up 5) – doesn't match.
• For \( g(x)=(x - 5)^2+2 \), vertex is \((5, 2)\) (right 5, up 2) – doesn't match.
• For \( g(x)=(x - 2)^2+5 \), vertex is \((2, 5)\) (right 2, up 5) – which is a valid shift from \( (0,0) \).

Answer:

\( \boldsymbol{g(x)=(x - 2)^2+5} \) (corresponding to the option: \( \boldsymbol{\text{O } g(x)=(x - 2)^2+5} \))