Problem 69 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \sqrt{x+2}\) is the graph of \(f(x) = \sqrt{x}\) shifted 2 units to the left.

Step 1: Graph the Original Function

First, let's graph the original function \(f(x) = \sqrt{x}\). This is a basic square root function where the value of x must be greater than or equal to 0. The graph starts from the origin (0,0) and increase as x increases, forming a curve in the first quadrant.

Step 2: Identify the Transformation

The given function \(g(x) = \sqrt{x+2}\) is a transformation of the original function \(f(x) = \sqrt{x}\). Specifically, it is a horizontal shift of the graph of the original function towards the left by 2 units. This is inferred from the 'x+2' under the square root, where the '+2' indicates a shift of 2 units to the left, opposite to the sign.

Step 3: Graph the Transformed Function

Now, plot the graph of the transformation \(g(x) = \sqrt{x+2}\), keeping in mind that the original curve must be moved 2 units to the left. This simply means all original x-values will be reduced by 2. So now the graph will start from (-2,0), instead of the origin (0,0), and increase as the x-values increase.

Key Concepts

Transformations of Functions

Graphs of mathematical functions can undergo various transformations that change their appearance without altering their original properties. A transformation can involve shifting, stretching, compressing, or reflecting the graph in the coordinate plane. These transformations provide a systematic way to manipulate the graph of a basic function, such as a square root, linear, quadratic, or absolute value function, to obtain the graph of a more complex function.
Understanding these transformations allows us to quickly sketch graphs of new functions based on the graphs of more familiar ones. For instance, the function transformation can be a simple shift to the left or right (horizontal shift), up or down (vertical shift), or more complex changes like stretching or shrinking the graph either vertically or horizontally. There are also reflections over the x-axis or y-axis, which essentially 'mirror' the graph across these axes.
In the example of graphing the square root function, the basic function \(f(x) = \sqrt{x}\) can be thought of as the 'parent' graph. By applying various transformations to this parent function, we create 'offspring' graphs that retain the shape of the original but are shifted, reflected, stretched, or compressed.

Horizontal Shift

A horizontal shift involves moving the graph of a function to the left or right along the x-axis. It is one of the simplest forms of transformation and significantly changes the domain of the function. To visualize a horizontal shift, imagine sliding the entire graph along the x-axis without altering its shape.
For the square root function \(f(x) = \sqrt{x}\), a horizontal shift can be represented in the form of \(f(x) = \sqrt{x - h}\) for a shift to the right by \(h\) units, or \(f(x) = \sqrt{x + h}\) for a shift to the left by \(h\) units. The direction of the shift is counterintuitive to some; adding a positive number within the function actually shifts the graph to the left, while subtracting moves it to the right. This confusion often stems from the fact that the function's input value is being increased or decreased before the function is applied.
In our example, the function \(g(x) = \sqrt{x+2}\) has a horizontal shift of 2 units to the left from the original function's position, due to the '+2' within the square root. The transformation shifted the starting point of the graph from \(x = 0\) to \(x = -2\), effectively moving the entire curve leftward.

Plotting Square Root Graphs

Plotting the graph of a square root function like \(f(x) = \sqrt{x}\) reveals its characteristic 'half-parabola' shape, opening to the right. When sketching square root graphs, keep in mind that the domain is restricted to \(x \geq 0\), because the square root of a negative number is not a real number.
To correctly plot a transformed square root graph, such as \(g(x) = \sqrt{x+2}\), begin by recognizing the starting point, which, for the basic square root function, is at the origin (0,0). Next, identify any transformations that have been applied—such as a horizontal shift as seen in our example—and adjust the starting point accordingly. For \(g(x)\), we move the starting point 2 units left to (-2,0). It's crucial to maintain the original shape of the graph; only the location of the starting point and subsequent points change.

• Plot the origin or shifted starting point.
• Select additional points that fit the function and mark their shifted locations.
  
• Connect the plotted points with a smooth curve that represents the shape of the square root function.
  
• Ensure the graph follows the natural increase of the square root function, where the rate of increase gradually decreases as the x-values grow.

To facilitate understanding, it can be helpful to create a table of values to plot specific points on the graph for both the original and transformed functions. This helps in keeping track of the horizontal shift and provides a clear visual of where to place each point on the coordinate plane.