Question

which function represents a translation of the parent cubic function 8 units to the left?
\\( h(x) = x^3 + 8 \\)
\\( h(x) = x^3 - 8 \\)
\\( h(x) = (x + 8)^3 \\)
\\( h(x) = (x - 8)^3 \\)

Explanation:

Step1: Recall horizontal translation rule

For a function \( y = f(x) \), a horizontal translation \( h \) units to the left is given by \( y = f(x + h) \), and \( h \) units to the right is \( y = f(x - h) \). The parent cubic function is \( f(x)=x^{3} \).

Step2: Apply the rule for 8 units left

We need to translate the parent cubic function \( f(x) = x^{3} \) 8 units to the left. Using the horizontal translation rule, replacing \( x \) with \( x + 8 \) in the parent function, we get \( h(x)=(x + 8)^{3} \).

Answer:

\( h(x)=(x + 8)^{3} \) (corresponding to the option \( h(x)=(x + 8)^{3} \))