Question

identify the graph for the equation.
y - 3 = \sqrt{x + 4}

Explanation:

Step1: Analyze the domain of the square root function

The expression inside the square root, \(x + 4\), must be non - negative. So we set up the inequality \(x+4\geq0\), which gives \(x\geq - 4\). This means the graph of the function should start at \(x=-4\) or to the right of \(x = - 4\).

Step2: Analyze the range of the function

We can rewrite the equation as \(y=3 + \sqrt{x + 4}\). Since the square root function \(\sqrt{x+4}\geq0\) for all \(x\) in its domain, then \(y=3+\sqrt{x + 4}\geq3\).

Step3: Analyze the three graphs

• For the first graph: The \(y\) - values are negative or close to zero, and it does not satisfy \(y\geq3\). Also, the domain seems to start at a positive \(x\) - value, not \(x=-4\).
• For the second graph: The \(y\) - values are negative or close to zero, and it does not satisfy \(y\geq3\).
• For the third graph: The domain starts around \(x=-4\) (or to the right of \(x = - 4\)) and the \(y\) - values are greater than or equal to \(3\), which matches our analysis of the domain and range of the function \(y = 3+\sqrt{x + 4}\).

Answer:

The third graph (the one with the \(y\) - axis labeled with values like 2, 4, 6, 8 and \(x\) - axis labeled with - 8, - 6, - 4, - 2) is the graph of the equation \(y - 3=\sqrt{x + 4}\).