Question

what value represents the horizontal translation from the graph of the parent function $f(x) = x^2$ to the graph of the function $g(x)=(x - 4)^2 + 2$?

• $-4$
• $-2$
• $2$
• $4$

Explanation:

Step1: Recall the vertex form of a parabola

The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For horizontal translations, the value of \( h \) determines the horizontal shift. If we have the parent function \( f(x)=x^2 \) (which has \( h = 0 \) and \( k = 0 \)) and the transformed function \( g(x)=(x - 4)^2+2 \), we compare it to the vertex form \( y=a(x - h)^2 + k \).

Step2: Identify the horizontal shift

In the function \( g(x)=(x - 4)^2+2 \), comparing with \( y=a(x - h)^2 + k \), we see that \( h = 4 \). The parent function \( f(x)=x^2 \) has its vertex at \((0,0)\), and the function \( g(x)\) has its vertex at \((4,2)\). The horizontal translation is determined by the change in the \( x \)-coordinate of the vertex. So the horizontal translation from \( f(x) \) to \( g(x) \) is \( 4 \) units to the right (since \( h = 4 \) and for the parent function \( h = 0 \), the shift is \( 4-0=4 \)).

Answer:

4 (the option corresponding to 4, which is the last option in the given choices)