Question

12 write the radical function represented by the graph:

Explanation:

Step1: Identify the parent function and transformations

The graph appears to be a square root function (radical function) with a horizontal shift. The parent function of a square root function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and starts at the origin \((0,0)\). Looking at the graph, the starting point (vertex) is at \((-6, -1)\)? Wait, no, let's check the coordinates. Wait, the grid: let's find the vertex. The black dot is at \((-5, -1)\)? Wait, maybe I misread. Wait, the x - axis: let's see the vertex of the radical graph. The standard square root function \( y=\sqrt{x} \) has vertex at \((0,0)\) and increases. This graph seems to be a transformation. Let's find the vertex. From the graph, the vertex (the starting point of the radical curve) is at \((h,k)\). Let's assume the vertex is at \((-6, -1)\)? Wait, no, let's check the grid. Let's count the units. Let's see, the vertex (the point where the curve starts) is at \((-6, -1)\)? Wait, maybe the vertex is at \((-6, -1)\), and the function is a square root function shifted horizontally and vertically. The general form of a square root function is \( y = a\sqrt{x - h}+k \), where \((h,k)\) is the vertex.

Wait, let's look at the graph again. The curve starts at a point, let's say the vertex is at \((-6, -1)\)? Wait, no, maybe the vertex is at \((-6, -1)\), and when \( x=-6 \), \( y = -1 \), and as \( x \) increases, \( y \) increases. Let's check a point. Let's take a point on the graph. Suppose when \( x = 0 \), what's \( y \)? Wait, maybe the vertex is at \((-6, -1)\), so \( h=-6 \), \( k = -1 \). Let's assume \( a = 1 \) (since the slope of the curve seems similar to the parent square root function). So the function would be \( y=\sqrt{x + 6}-1 \). Wait, let's verify. When \( x=-6 \), \( y=\sqrt{-6 + 6}-1=\sqrt{0}-1=-1 \), which matches the vertex. Now, let's check another point. Let's take \( x = 3 \) (since from \( x=-6 \), moving 9 units to the right, \( x=-6 + 9=3 \)), then \( y=\sqrt{3 + 6}-1=\sqrt{9}-1=3 - 1 = 2 \). Wait, does the graph pass through \((3,2)\)? Let's see the grid. If the vertex is at \((-6, -1)\), and when \( x = 3 \), \( y = 2 \), that seems to fit. Alternatively, maybe the vertex is at \((-5, -1)\). Wait, maybe I made a mistake. Let's re - examine.

Wait, the black dot is at \((-5, -1)\)? Wait, no, the x - axis labels: let's see, the x - axis has numbers like - 10, - 9, - 8, - 7, - 6, - 5, - 4, etc. The vertex of the radical curve (the point where the curve starts) is at \((-6, -1)\)? Wait, maybe the correct vertex is \((-6, -1)\). So the general form is \( y=\sqrt{x - h}+k \), with \( h=-6 \), \( k=-1 \), so \( y=\sqrt{x + 6}-1 \). Let's check the domain: \( x+6\geq0\Rightarrow x\geq - 6 \), which matches the graph (the curve starts at \( x=-6 \) and goes to the right).

Step2: Confirm the transformation

The parent function \( y = \sqrt{x} \) has a vertex at \((0,0)\), domain \( x\geq0 \), range \( y\geq0 \). Our function \( y=\sqrt{x + 6}-1 \) has a vertex at \((-6, -1)\), domain \( x\geq - 6 \), range \( y\geq - 1 \), which matches the graph's starting point and direction (increasing as \( x \) increases).

Answer:

The radical function represented by the graph is \( y=\sqrt{x + 6}-1 \)