Question

graph the following function
$y = \sqrt{x + 3}$
choose the best graph

Explanation:

Step1: Find the domain

For \( y = \sqrt{x + 3} \), the expression under the square root must be non - negative. So, \( x+3\geq0 \), which gives \( x\geq - 3 \).

Step2: Find the y - intercept

To find the y - intercept, set \( x = 0 \). Then \( y=\sqrt{0 + 3}=\sqrt{3}\approx1.732 \).

Step3: Analyze the shape

The parent function is \( y=\sqrt{x} \), which has a domain \( x\geq0 \) and starts at the point \( (0,0) \) and increases slowly. The function \( y=\sqrt{x + 3} \) is a horizontal shift of the parent function \( y = \sqrt{x} \) 3 units to the left. So, its graph should start at the point \( (-3,0) \) (since when \( x=-3 \), \( y = 0 \)) and increase. Also, when \( x = 0 \), \( y=\sqrt{3}\approx1.732 \), which is a positive value less than 2.

Looking at the options:

• Option A: The graph seems to start at \( x = 0 \) (the vertex is at \( x = 0 \)), which is incorrect as our function starts at \( x=-3 \).
• Option B: The graph has a very small slope and does not match the expected value of \( y\) at \( x = 0 \).
• Option C: The graph starts at \( x=-3 \) (since the vertex is at \( x=-3 \)) and when \( x = 0 \), the \( y\) - value is around \( \sqrt{3}\approx1.732 \), which matches our calculations.
• Option D: The graph starts at a positive \( x\) - value, which is incorrect as the domain starts at \( x=-3 \).

Answer:

C