Problem 53 - Precalculus 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 \(f\). $$\begin{aligned} &f(x)=\sqrt{x} \quad(x=0,1,4,9) \text { and }\\\ &g(x)=\sqrt{x-1} \quad(x=1,2,5,10) \end{aligned}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = \sqrt{x-1}\) is a horizontal translation of the graph of \(f(x) = \sqrt{x}\), shifted one unit to the right.

Step 1: Determine the points for the graph of f

For the function \(f(x)=\sqrt{x}\), the given values of x are 0, 1, 4, and 9. Substituting these values into the equation we obtain: \(f(0)=\sqrt{0}=0\), \(f(1)=\sqrt{1}=1\), \(f(4)=\sqrt{4}=2\), \(f(9)=\sqrt{9}=3\). So, the points to plot for \(f(x)\) are (0,0), (1,1), (4,2), and (9,3).

Step 2: Determine the points for the graph of g

For the function \(g(x)=\sqrt{x-1}\), the given values of x are 1, 2, 5, and 10. Substituting these values into the equation we obtain: \(g(1)=\sqrt{1-1}=0\), \(g(2)=\sqrt{2-1}=1\), \(g(5)=\sqrt{5-1}=2\), \(g(10)=\sqrt{10-1}=3\). So, the points to plot for \(g(x)\) are (1,0), (2,1), (5,2), and (10,3).

Step 3: Draw the Graph

Plot the points on the same rectangular coordinate system and connect them with a smooth curve. Graph the curve of \(f(x)=\sqrt{x}\) starting from (0,0), and the curve of \(g(x)=\sqrt{x-1}\) starting from (1,0). Both curves should only appear for nonnegative values of x.

Step 4: Analyze the Relation between the two graphs

The two graphs are similar in shape, but the graph of \(g(x) = \sqrt{x-1}\) appears to be shifted one unit to the right in comparison to the graph of \(f(x) = \sqrt{x}\). Therefore, the graph of \(g\) is a horizontal translation of the graph of \(f\) by one unit to the right.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundational tool in graphing functions. It consists of two perpendicular lines called axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, designated by the coordinates (0,0).

To graph a function, we determine points, which are pairs of x and y values known as ordered pairs, and plot them on the coordinate system. Each point represents an input-output relationship within the function - the x value denotes the input, and the corresponding y value denotes the output.

Radical Expressions

Radical expressions include numbers or expressions under the root symbol. In the case of square root functions like the ones in our example, the expression under the radical tells us which numbers we can plug into the function. Since we can only take the square root of nonnegative numbers to get real number results, this constrains the domain (the allowable x-values) of the function.

For instance, the expression \(\sqrt{x}\) can take any nonnegative x-value, whereas \(\sqrt{x-1}\) can only take x-values greater than or equal to 1. This is how we ensure that the expressions under the radical sign are nonnegative.

Horizontal Translation

When we talk about horizontal translation in the context of function transformation, we are referring to the shifting of the entire graph of a function left or right along the x-axis.

This shift is determined by an addition or subtraction within the function's formula. In our example, the function \(g(x) = \sqrt{x-1}\) represents a horizontal translation of the function \(f(x) = \sqrt{x}\) by 1 unit to the right because of the subtraction by 1 inside the radical. The graph maintains its shape but is relocated along the x-axis.

Ordered Pairs

Ordered pairs are the building blocks for graphing functions on a coordinate plane. Each ordered pair \( (x, y) \) relates the input x to the output y of a function. To graph the square root function \(f(x)\), we calculate y by applying the function to x and thereby obtain ordered pairs which we then plot on the graph.

In our exercise, the substitution process for the function \(f(x)\) and \(g(x)\) yields specific ordered pairs that lay out the path of the graph across the plane. For instance, for \(f(x) = \sqrt{x}\), we substitute x with 0, 1, 4, and 9 to get the pairs (0,0), (1,1), (4,2), and (9,3) respectively.

Function Transformation

Function transformation encompasses a variety of changes applied to a function's graph, such as shifting, stretching, compressing, and reflecting. These transformations alter the appearance of the original graph but do not always change its fundamental shape.

In our example, the square root function \(g(x)\) is derived from \(f(x)\) by applying a horizontal translation, which is a specific type of function transformation. Recognizing transformations helps us understand how different functions relate to one another and predict the appearance of their graphs.