Question

identify the graph of ( h(x) = (x - 2)^2 + 4 )
images of graphs
compare the graph to the graph of ( f(x) = x^2 ).
the graph of ( h ) is a translation (\boxed{}) and (\boxed{}) of the graph of ( f ).

Explanation:

Step1: Recall the vertex form of a parabola

The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. For \(h(x)=(x - 2)^2+4\), comparing with \(y = a(x - h)^2 + k\), we have \(h = 2\) and \(k = 4\). For the parent function \(f(x)=x^2\), its vertex is \((0,0)\).

Step2: Determine the horizontal translation

The horizontal shift is determined by the value of \(h\). If \(h>0\), the graph shifts \(h\) units to the right; if \(h < 0\), it shifts \(|h|\) units to the left. Here \(h = 2>0\), so the graph of \(h(x)\) is a horizontal translation of \(f(x)\) 2 units to the right.

Step3: Determine the vertical translation

The vertical shift is determined by the value of \(k\). If \(k>0\), the graph shifts \(k\) units up; if \(k < 0\), it shifts \(|k|\) units down. Here \(k = 4>0\), so the graph of \(h(x)\) is a vertical translation of \(f(x)\) 4 units up.

Answer:

The graph of \(h\) is a translation \(\boldsymbol{2}\) units to the right and \(\boldsymbol{4}\) units up of the graph of \(f\).