Question

7 rewrite $y = \sqrt{4x + 16} + 5$ to make it easy to graph using a translation. describe the graph.

• the equation is $y = 2\sqrt{x + 4} + 5$. it is the graph of $y = 2\sqrt{x}$ translated 4 units left and 5 units up.
• the equation is $y = \sqrt{x + 4} + 5$. it is the graph of $y = \sqrt{x}$ translated 4 units right and 5 units up.
• the equation is $y = 2\sqrt{x + 4} + 5$. it is the graph of $y = 2\sqrt{x}$ translated 4 units right and 5 units up.
• the equation is $y = \sqrt{x - 4} + 5$. it is the graph of $y = \sqrt{x}$ translated 4 units left and 5 units up.

Explanation:

1. First, simplify the original function \( y=\sqrt{4x + 16}+5 \). Factor out 4 from the square - root: \( \sqrt{4(x + 4)}+5 \). Since \( \sqrt{4(x + 4)}=\sqrt{4}\times\sqrt{x + 4}=2\sqrt{x + 4} \), the function simplifies to \( y = 2\sqrt{x + 4}+5 \).
2. Recall the transformation rules of functions: For a function \( y = f(x) \), the function \( y=f(x + h)+k \) is the graph of \( y = f(x) \) translated \( h \) units to the left (if \( h>0 \)) and \( k \) units up (if \( k > 0 \)). Here, our base function is \( y = 2\sqrt{x} \), and the transformed function is \( y=2\sqrt{x+4}+5 \), where \( h = 4 \) (so 4 units left) and \( k = 5 \) (so 5 units up).
3. Analyze the other options:

• The second option has an incorrect simplification (it should have a factor of 2 from the square - root of 4).
• The third option has an incorrect direction of translation ( \( x+4 \) indicates a left translation, not a right translation).
• The fourth option has an incorrect simplification (the square - root simplification is wrong and the translation direction is wrong).

Answer:

The first option (the one with "The equation is \( y = 2\sqrt{x + 4} + 5 \). It is the graph of \( y = 2\sqrt{x} \) translated 4 units left and 5 units up")