Question

guided practice graphing radical functions which function describes the graph? a. $y = \sqrt{x + 5} + 5$ b. $y = \sqrt{x - 5} + 5$ c. $y = \sqrt{x - 5} - 5$ d. $y = \sqrt{x + 5} - 5$ (graph shows a coordinate plane with a curve starting near (-6,6) and moving towards the right, with x-axis labeled from -6 to 10 and y-axis labeled from -2 to 10)

Explanation:

Step1: Identify the vertex of the graph

The graph of the radical function (square root function) has its starting point (vertex) at \( (-5, -5) \) (by looking at the grid: x - coordinate is - 6? Wait, no, let's check the graph. Wait, the x - axis has marks at - 6, - 4, - 2, 0, 2,... and y - axis at - 6, - 4, - 2, 0, 2,... Wait, the graph starts around x=-5 (since between - 6 and - 4, closer to - 6? Wait, no, the standard form of a square root function is \( y=\sqrt{x - h}+k \), where \( (h,k) \) is the vertex. Wait, no, the parent function is \( y = \sqrt{x} \), which has vertex at \( (0,0) \). For a transformation \( y=\sqrt{x - h}+k \), the vertex is \( (h,k) \). Wait, no, actually, \( y=\sqrt{x + a}+b=\sqrt{x-(-a)}+b \), so the vertex is \( (-a,b) \).

Looking at the graph, the vertex (the starting point of the square root curve) seems to be at \( (-5, - 5) \).

Step2: Match with the function form

The general form of a square root function is \( y=\sqrt{x - h}+k \), where \( (h,k) \) is the vertex. Wait, no, if the vertex is \( (-5,-5) \), then \( h=-5 \) and \( k = - 5 \)? Wait, no, let's re - express: \( y=\sqrt{x - h}+k \), vertex at \( (h,k) \). So if vertex is \( (-5,-5) \), then \( h=-5 \) and \( k=-5 \). So substituting into the formula: \( y=\sqrt{x-(-5)}+(-5)=\sqrt{x + 5}-5 \), which is option D.

Let's check other options:

• Option A: \( y=\sqrt{x + 5}+5 \), vertex at \( (-5,5) \), which is not matching the graph's vertex (which is at a negative y - value).
• Option B: \( y=\sqrt{x - 5}+5 \), vertex at \( (5,5) \), x - coordinate is positive, but our graph starts at negative x, so eliminate.
• Option C: \( y=\sqrt{x - 5}-5 \), vertex at \( (5,-5) \), x - coordinate positive, eliminate.
• Option D: \( y=\sqrt{x + 5}-5 \), vertex at \( (-5,-5) \), which matches the vertex we identified from the graph.

Answer:

D. \( y = \sqrt{x + 5} - 5 \)