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 translation, we look at the value of \( h \).

Step2: Compare with the given function

The parent function is \( f(x)=x^2 \) (which can be written as \( f(x) = 1(x - 0)^2 + 0 \)), and the given function is \( g(x)=(x - 4)^2 + 2 \). Comparing with the vertex form \( y = a(x - h)^2 + k \), here \( h = 4 \). In terms of horizontal translation from the parent function \( f(x)=x^2 \) (where \( h = 0 \)) to \( g(x) \) (where \( h = 4 \)), the horizontal translation is determined by the change in \( h \). The rule for horizontal translation in the vertex form \( y=(x - h)^2 \) is that if \( h>0 \), the graph shifts \( h \) units to the right, and if \( h < 0 \), it shifts \(|h|\) units to the left. Here, since \( h = 4 \) in \( g(x)=(x - 4)^2+2 \), the horizontal translation from \( f(x)=x^2 \) (where \( h = 0 \)) is 4 units to the right. So the value representing the horizontal translation is 4.

Answer:

4 (corresponding to the option "4")