Question

given g(x)=\frac{1}{8}(x + 2)^2 + 6, identify the parent function and transformations
circle all that apply:
inear
bsolute value
orizontal translation 2 units right
izontal translation 2 units left
quadratic
exponential
vertical translation 6
vertical tra
vertical stretc

Explanation:

Step1: Identify Parent Function

The given function is \( g(x)=\frac{1}{8}(x + 2)^2+6 \). The general form of a quadratic function is \( f(x)=a(x - h)^2 + k \), where \( a\), \(h\), \(k\) are constants. So the parent function is the quadratic function \( f(x)=x^2 \).

Step2: Analyze Horizontal Translation

In the form \( a(x - h)^2 + k \), the horizontal shift is determined by \( h \). Here, the function is \( \frac{1}{8}(x - (-2))^2+6 \), so \( h=-2 \). A horizontal translation of \( |h| \) units: if \( h<0 \), it is a shift to the left. Since \( h = - 2\), the horizontal translation is 2 units to the left.

Step3: Analyze Vertical Translation

The vertical shift is determined by \( k \). Here \( k = 6\), so there is a vertical translation of 6 units up.

Step4: Analyze Vertical Stretch/Compression

The coefficient \( a=\frac{1}{8} \). Since \( 0<|a|<1 \), it is a vertical compression (or a vertical stretch by a factor less than 1) by a factor of \( \frac{1}{8} \).

Answer:

Parent function: Quadratic (\( f(x)=x^2 \))
Transformations: Horizontal Translation 2 units left, Vertical Translation 6 units up, Vertical Compression (by a factor of \( \frac{1}{8} \))