Problem 72 - 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 \(h(x)=-\sqrt{x+1}\) is a reflection of the graph of the original square root function \(f(x)=\sqrt{x}\) that has been shifted 1 unit to the left, resulting in a shape that looks like a half-parabola opening to the left and decreasing along the x-axis.

Step 1: Graph the Basic Square Root Function

We begin by graphing the basic square root function \(f(x)=\sqrt{x}\). This is a half-parabola shape opening to the right, starting at the origin (0,0) and increasing as we move to the right along the x-axis.

Step 2: Apply Horizontal Shift

Next, we apply the horizontal shift. In the function \(h(x)=-\sqrt{x+1}\), the term inside the square root, \(x+1\), results in a shift of the graph 1 unit to the left. This is equivalent to replacing every x-coordinate in the original graph of \(f(x)=\sqrt{x}\) by x-1. So the graph of \(f(x)=\sqrt{x}\) shifted one unit to the left would look like the graph of \(f(x)=\sqrt{x+1}\) before applying the negative sign.

Step 3: Apply Reflection Across the X-Axis

Lastly, the negative sign in front of the square root in \(h(x)=-\sqrt{x+1}\) flips the graph across the x-axis. The y-coordinates of all points on the shifted graph of \(f(x)=\sqrt{x+1}\) are multiplied by -1 to reflect about the x-axis, resulting in the graph of \(h(x)=-\sqrt{x+1}\).

Step 4: Complete Graph

Combine the results from steps 2 and 3 to complete the graph of the given function \(h(x)=-\sqrt{x+1}\), which has been shifted 1 unit to the left from the original square root function and reflected across the x-axis.