Question

question 3
1 pts
which transformations have been performed on the graph of $f(x) = \sqrt{x}$ to obtain the graph of $g(x) = \frac{1}{4}\sqrt{x + 3} + 6$ ?
select each correct answer.

• compress the graph closer to the x-axis
• stretch the graph away from the x-axis
• translate the graph to the right
• translate the graph down
• reflect the graph over the x-axis
• translate the graph to the left
• translate the graph up

Explanation:

Step1: Analyze Vertical Compression/Stretch

For a function \( y = a\sqrt{x} \), if \( 0 < |a| < 1 \), it's a vertical compression. Here, \( a=\frac{1}{4} \), so the graph is compressed closer to the x - axis.

Step2: Analyze Horizontal Translation

For a function \( y=\sqrt{x + h} \), if \( h>0 \), it's a translation to the left by \( h \) units. Here, \( h = 3 \), so the graph is translated 3 units to the left.

Step3: Analyze Vertical Translation

For a function \( y=\sqrt{x}+k \), if \( k>0 \), it's a translation up by \( k \) units. Here, \( k = 6 \), so the graph is translated 6 units up.

Answer:

• compress the graph closer to the x - axis
• translate the graph to the left
• translate the graph up