Problem 68 - 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. $$g(x)=\sqrt{x}+1$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\sqrt{x}+1\) is a half parabola beginning at the point (0,1) and expanding towards the positive x-axis. It is similar to the base function \(f(x)=\sqrt{x}\), but is shifted upward by one unit.

Step 1: Graph the Base Function \(f(x)=\sqrt{x}\)

Start by plotting the square root function. This function forms a half parabola that begins at the origin (0,0) and expands towards the right or positive x-axis. Prevalent points on this graph are (0,0), (1,1), (4,2), and (9,3). The shape of the graph is determined by the fact that the square root of x increases as x increases, but it does so at a decelerating rate. Add these points to the graph and connect them smoothly for the visual illustration of \(f(x)=\sqrt{x}\)

Step 2: Understand the Transformation

Next, look at the transformation that turns \(f(x)\) into \(g(x)\). Given that \(g(x)=\sqrt{x}+1\), this represents a vertical shifting of the original function \(f(x)\) upwards by one unit. This means the entire graph of \(f(x)\) will be shifted up by a single unit. This is a simple transformation that does not alter the overall shape of the graph

Step 3: Graph the Transformed Function \(g(x) = \sqrt{x}+1\)

To graph \(g(x)=\sqrt{x}+1\), move every point on the original graph (the graph of \(f(x)=\sqrt{x}\)) up by one unit. This results in the points (0,1), (1,2), (4,3), and (9,4). Plot these points and connect them smoothly to get the graph of \(g(x)=\sqrt{x}+1\). This, like \(f(x)\), will be a half parabola. However, it will start at (0,1) instead of the origin and expands towards the positive x-axis from there.