Question

for the following expression, $2f(x + 3) + 7$, write this as a square root function and state the transformations
function: $g(x) = \frac{1}{2}\sqrt{x + 3} + 7$
transformations:
options:
$g(x) = 2\sqrt{x - 3} + 7$; vertical compression by 2; vertical translation down 7; $g(x) = 2\sqrt{x + 3} + 7$
vertical stretch by 2; horizontal translation right 3; horizontal translation left 3; vertical translation up 7
horizontal translation left 7

Explanation:

Step1: Identify the parent function

The parent square root function is \( f(x)=\sqrt{x} \). We need to transform this to \( 2f(x + 3)+7 \).

Step2: Write the function

Substitute \( f(x)=\sqrt{x} \) into \( 2f(x + 3)+7 \). So \( f(x + 3)=\sqrt{x + 3} \), then \( 2f(x + 3)+7 = 2\sqrt{x + 3}+7 \). So the correct function is \( g(x)=2\sqrt{x + 3}+7 \).

Step3: Analyze transformations

• Horizontal translation: For a function \( f(x + h) \), if \( h>0 \), it's a horizontal translation left by \( h \) units. Here we have \( x+3 \), so \( h = 3 \), which is a horizontal translation left 3 units.
• Vertical stretch/compression: The coefficient \( 2 \) in front of \( \sqrt{x + 3} \) is a vertical stretch by a factor of \( 2 \) (since \( 2>1 \), it's a stretch, not a compression).
• Vertical translation: The \( +7 \) at the end is a vertical translation up 7 units (since we add 7 to the function value).

Answer:

Function: \( \boldsymbol{g(x)=2\sqrt{x + 3}+7} \)

Transformations: Vertical Stretch by 2, Horizontal translation left 3, vertical translation up 7