Problem 74 - 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. $$h(x)=\sqrt{-x+1}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x)= \sqrt{-x+1}\) is a reflection of the basic square root function \(f(x) = \sqrt{x}\) about the y-axis that is then shifted right by one unit.

Step 1: Graph the Basic Square Root Function

Begin by plotting the basic square root function, \(f(x) = \sqrt{x}\). Draw the x-axis and y-axis. Indicate some key points such as (0,0), (1,1), (4,2), and (9,3). Plot these points on the graph and carefully sketch the curve that represents the square root function. The graph starts from the origin and extends to the right, curving upward.

Step 2: Reflect the Basic Function

The negative sign before the \(x\) within the square root in \(h(x) = \sqrt{-x + 1}\) signifies a reflection of the function about the y-axis. Reflect the graph of the basic function plotted earlier along the y-axis. Now, instead of curving upward to the right, the function curves upward to the left.

Step 3: Shift the Reflected Function

The term \(+1\) within the square root represents a horizontal shift of the graph of one unit to the right. Shift the reflected graph one unit to the right. There you have the graph of the function \(h(x) = \sqrt{-x + 1}\).

Step 4: Label the Graph

Ensure that the new graph is labeled correctly. The graph should start at the point (1,0) and curve upward to the left, and the curve should be similar to the initial shape of the base square root function.