1. **State the problem:** We need to identify which equation matches the given graph, which is a blue curve in the top-left quadrant starting near (-2, 0) and curving upward to the right toward (6, 2). 2. **Recall the general form of a square root function:** $$y = \sqrt{x}$$ Shifts in the graph correspond to changes inside the square root (horizontal shifts) and outside (vertical shifts). 3. **Analyze each equation:** - Equation 1: $$y - 1 = \sqrt{x + 3} \implies y = 1 + \sqrt{x + 3}$$ - Domain: $$x + 3 \geq 0 \Rightarrow x \geq -3$$ - Starts at $$x = -3$$, $$y = 1 + 0 = 1$$ - Equation 2: $$y + 3 = \sqrt{x - 1} \implies y = -3 + \sqrt{x - 1}$$ - Domain: $$x - 1 \geq 0 \Rightarrow x \geq 1$$ - Starts at $$x = 1$$, $$y = -3 + 0 = -3$$ - Equation 3: $$y + 1 = \sqrt{x - 3} \implies y = -1 + \sqrt{x - 3}$$ - Domain: $$x - 3 \geq 0 \Rightarrow x \geq 3$$ - Starts at $$x = 3$$, $$y = -1 + 0 = -1$$ 4. **Compare with the graph:** - The graph starts near (-2, 0), so the domain should include $$x = -2$$. - Equation 1's domain $$x \geq -3$$ includes $$-2$$, and at $$x = -2$$: $$y = 1 + \sqrt{-2 + 3} = 1 + \sqrt{1} = 1 + 1 = 2$$ - Equation 2's domain $$x \geq 1$$ does not include $$-2$$. - Equation 3's domain $$x \geq 3$$ does not include $$-2$$. 5. **Check the shape and position:** - Equation 1 starts near (-3,1) and increases as $$x$$ increases, matching the graph's upward curve from left to right. **Final answer:** The equation matching the graph is $$y - 1 = \sqrt{x + 3}$$