Problem 73 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\sqrt{-x+2}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = \sqrt{-x+2}\) is the graph of the square root function \(f(x) = \sqrt{x}\) reflected on the y-axis and shifted 2 units to the right.

Step 1: Graph the Base Function

Draw the graph of the function \(f(x) = \sqrt{x}\). This function represents a half-parabola, starting from point (0, 0) and rising to the right.

Step 2: Understand the Transformation

The given function is \(h(x) = \sqrt{-x+2}\). Here, two transformations are taking place. The negative sign before \(x\) inside the square root indicates a reflection on the vertical line, while '+2' inside the square root indicates a horizontal shift to the right by 2 units.

Step 3: Apply the Transformation

First reflect the original graph of the function \(f(x) = \sqrt{x}\) about the y-axis because of the negative sign. Then, shift the reflected graph to the right by 2 units due to '+2' inside the square root. This is your graph of function \(h(x) = \sqrt{-x+2}\).

Step 4: Check Your Solution

Make sure that the new graph is a reflection of the original square root function and that it has been correctly shifted 2 units to the right.