Question

which best describes the transformation that occurs from the graph of $f(x) = x^2$ to $g(x) = (x - 2)^2 + 3$? \bigcirc right 2, up 3 \bigcirc left 2, down 3 \bigcirc right 2, down 3 \bigcirc left 2, up 3

Explanation:

Step1: Recall transformation rules

For a quadratic function in the form \( y = (x - h)^2 + k \), the vertex is at \((h, k)\). The parent function \( f(x)=x^2 \) has vertex \((0, 0)\). For \( g(x)=(x - 2)^2 + 3 \), the vertex is \((2, 3)\).

Step2: Analyze horizontal shift

The horizontal shift is determined by \( h \). If \( h>0 \), the graph shifts right by \( h \) units. Here, \( h = 2 \), so the graph shifts right 2 units.

Step3: Analyze vertical shift

The vertical shift is determined by \( k \). If \( k>0 \), the graph shifts up by \( k \) units. Here, \( k = 3 \), so the graph shifts up 3 units.

Answer:

right 2, up 3