Question

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Explanation:

Step1: Analyze the reflection

The function \( n(x) = - (x + 4)^2 + 2 \) and the parent function \( f(x)=x^2 \). The negative sign in front of \( (x + 4)^2 \) means the graph of \( n(x) \) is a reflection of \( f(x) \) over the \( x \)-axis (since for \( y = -f(x) \), it's a reflection over \( x \)-axis).

Step2: Analyze horizontal translation

For a quadratic function in the form \( y = a(x - h)^2 + k \), the horizontal shift is determined by \( h \). Here, \( n(x)=-(x - (- 4))^2+2 \), so compared to \( f(x)=x^2 \) (where \( h = 0 \)), the graph of \( n(x) \) is shifted 4 units left (because \( h=-4 \), and the formula for horizontal shift is: if \( h<0 \), shift \( |h| \) units left).

Step3: Analyze vertical translation

The vertical shift is determined by \( k \). In \( n(x)=-(x + 4)^2+2 \), \( k = 2 \), so compared to \( f(x)=x^2 \) (where \( k = 0 \)), the graph of \( n(x) \) is shifted 2 units up.

Answer:

The graph of \( n \) is a reflection over the \( x \)-axis and a translation 4 units left and 2 units up of the graph of \( f \). So the answers for the blanks (in order) are: over the \( x \)-axis, 4 units left, 2 units up.