Problem 53 - College Algebra Essentials

Question

Graph the given square root functions, \(f\) and \(g,\) in the same rectangular coordinate system. Use the integer values of \(x\) given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of \(x\) that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of \(f\). $$ \begin{array}{l} f(x)=\sqrt{x}(x=0,1,4,9) \text { and } \\ g(x)=\sqrt{x-1} \quad(x=1,2,5,10) \end{array} $$

Step-by-Step Solution

Answer

The graph of function \(g(x) = \sqrt{x-1}\) is a horizontal shift of the graph of function \(f(x) = \sqrt{x}\) 1 unit to the right.

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

The values of \(f\) for each \(x = 0, 1, 4, 9\) are \(0, 1, 2, 3\) respectively, which correspond to the points \((0, 0), (1, 1), (4, 2), (9, 3)\). Plot these points on the graph and connect them smoothly to draw the curve.

Step 2: Graph the function \(g(x) = \sqrt{x-1}\)

Now, calculate the values of \(g\) for each \(x = 1, 2, 5, 10\). They are \(0, 1, 2, 3\) respectively, which correspond to the points \((1, 0), (2, 1), (5, 2), (10, 3)\). Plot these points on the same coordinate system and again connect them smoothly to create the curve.

Step 3: Comparing the graphs

After plotting both functions on the same coordinate system, look at how the two graphs relate to each other. One important thing to notice is that the graph of \(g(x) = \sqrt{x-1}\) is a horizontal shift of 1 unit to the right of the graph of \(f(x) = \sqrt{x}\).

Key Concepts

Coordinate System

Understanding the coordinate system is crucial for graphing square root functions effectively. The coordinate system is a two-dimensional plane composed of a horizontal axis, known as the x-axis, and a vertical axis, the y-axis. These axes intersect at a point called the origin, labeled as (0,0). Points are represented as ordered pairs

• The first value indicates the position along the x-axis
• The second value indicates the position along the y-axis

This system helps us plot graphs by determining each point’s position relative to these axes. When graphing functions, such as square root functions, you plot points generated by substituting x-values into the function and calculate corresponding y-values, which are then plotted as (x,y) on this system. This clear visualization is key to comparing different functions easily.

Function Transformation

Function transformation is an essential concept when dealing with graphing functions like square root functions. Transformation involves shifting, stretching, or shrinking the graph of a function to produce a new graph. In the context of our problem:

• We see a horizontal shift in the graph of the function \(g(x) = \sqrt{x-1}\)
• This transformation shifts the basic graph of \(f(x) = \sqrt{x}\), 1 unit to the right.

This horizontal shift occurs because the function is modified from \(\sqrt{x}\) to \(\sqrt{x-1}\). As a rule, subtracting inside the square root function moves the graph to the right. Such transformations help us understand how small changes to a function’s equation affect its graph, allowing us to manipulate and fine-tune functions for various applications.

Square Root Properties

Square root properties guide us in understanding and graphing square root functions effectively. A square root function is essentially defined as \(f(x) = \sqrt{x}\), where the output is the non-negative value, or principal square root, of x. There are key characteristics specific to these functions:

• Square roots are only defined for non-negative numbers, since negative roots are not real numbers.
• The graph of \(f(x) = \sqrt{x}\) begins at the origin (0,0) and rises slowly to the right, providing a visual representation that the function gradually increases.

Understanding these properties is crucial when choosing appropriate x-values, ensuring that the numbers result in a real y-value. This restriction is why in our problem, we only use x-values that keep the expression under the radical non-negative, ensuring all plotted points are real and valid on the coordinate system.