Problem 80 - 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. $$g(x)=2 \sqrt{x+1}-1$$

Step-by-Step Solution

Answer

The graph of \(g(x)=2 \sqrt{x+1}-1\) is a transformation of the base square root function. After a vertical stretch by factor of 2, it shifts left by 1 unit and down by 1 unit.

Step 1: Graphing the base function

Begin by sketching the graph of base function \(f(x)=\sqrt{x}\). This graph is a curve that starts from the origin (0,0) and extends upwards and towards the right.

Step 2: Applying vertical stretching

Now, apply the vertical stretching of the function by factor of 2. You accomplish this by simply multiplying the \(y\) values of the base function by 2. That is, multiply the square root function by 2.

Step 3: Horizontal shift

Next is the horizontal transformation. For \(g(x)=2 \sqrt{x+1}-1\), the graph is moved 1 unit to the left. This means that for every \(x\), consider the corresponding \(x-1\) for the new function.

Step 4: Vertical shift

Finally, comes the vertical transformation. The function is to be moved down by 1 unit, reflected in the -1 in the function \(g(x)=2 \sqrt{x+1}-1\). This means that for every \(y\), consider the corresponding \(y-1\) for the new function.

Step 5: Graphing the transformed function

With all transformations applied, you can now sketch the graph of the new function \(g(x)=2 \sqrt{x+1}-1\). Plot a set of points using the transformed function and then smooth the points with a curve.