Question

the graph above is a transformation of the function \\(\sqrt{x}\\)

write an equation for the function graphed above

\\(g(x) =\\)

Explanation:

Identify the starting point of the radical function

The standard square root function \(f(x) = \sqrt{x}\) starts at the origin \((0,0)\).
Looking at the given graph, the endpoint (starting point) of the curve is located at \((-2, 2)\).
This indicates a horizontal shift to the left by 2 units and a vertical shift up by 2 units.

Set up the general transformation equation

The general equation for a transformed square root function is:

$$g(x) = a\sqrt{x - h} + k$$

where \((h, k)\) is the starting point of the radical.
Substituting the starting point \((h, k) = (-2, 2)\):

$$g(x) = a\sqrt{x - (-2)} + 2$$

$$g(x) = a\sqrt{x + 2} + 2$$

Determine the vertical stretch factor

To find the value of the constant \(a\), we select another clear point on the graph.
Looking at the grid, the curve passes perfectly through the point \((2, 4)\).
Substitute \(x = 2\) and \(g(x) = 4\) into our equation:

$$4 = a\sqrt{2 + 2} + 2$$

$$4 = a\sqrt{4} + 2$$

$$4 = 2a + 2$$

Solve for the parameter

Subtract 2 from both sides:

$$2 = 2a$$

Divide by 2:

$$a = 1$$

Write the final function

Substituting \(a = 1\) back into the equation yields:

$$g(x) = \sqrt{x + 2} + 2$$

Answer:

Write an equation for the function graphed above

\(g(x) =\) <blank>\(\sqrt{x+2}+2\)</blank>