Problem 54 - 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 \(f\). $$ \begin{aligned} &f(x)=\sqrt{x} \quad(x=0,1,4,9) \text { and }\\\ &g(x)=\sqrt{x+2} \quad(x=-2,-1,2,7) \end{aligned} $$

Step-by-Step Solution

Answer

The graph of \(g(x) = \sqrt{x+2}\) is a horizontal shift 2 units to the left from the graph of \(f(x) = \sqrt{x}\). Both functions maintain their characteristic square root function shape.

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

Using the given values of \(x=0,1,4,9\), find the corresponding values of \(f(x)\) by substituting \(x\) into \(f(x)=\sqrt{x}\). These values give the coordinate points (0,0), (1,1), (4,2), (9,3). Now plot these points on the rectangular coordinate system.

Step 2: Plot function \(g(x) = \sqrt{x+2}\)

Using the given values of \(x=-2,-1,2,7\), find the corresponding values of \(g(x)\) by substituting \(x\) into \(g(x)=\sqrt{x+2}\). These values give the coordinate points (-2,0), (-1,1), (2,2), (7,3). Now plot these points on the same rectangular coordinate system.

Step 3: Analyze the relationship between the graphs of \(f\) and \(g\)

Observe the position and behavior of the two graphs relative to each other. One could observe that the graph of \(g(x)\) is the graph of \(f(x)\) shifted 2 units to the left. The square root functions maintain their characteristic shape with this horizontal shift.

Key Concepts

Graphing Functions

When graphing functions, especially square root functions, it's essential to follow some simple steps. Start by choosing specific values of \(x\) for which you'll find corresponding \(y\) values. In this exercise, we graph two functions: \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{x+2}\).

For \(f(x)\), we use \(x = 0, 1, 4,\) and \(9\), resulting in the points \((0,0), (1,1), (4,2),\) and \((9,3)\). These form a gentle curve upwards, typical of square root functions.

It's important to plot these points accurately within the coordinate system to visualize the function's unique shape correctly and understand the relationship between different functions.

Transformation of Functions

Transformation is a key concept in understanding how functions can change. Here, the transformation involves shifting \(f(x) = \sqrt{x}\) to form \(g(x) = \sqrt{x+2}\).

The transformation happens because \(g(x)\) is derived by taking the expression under the square root and adding 2. This causes the graph to shift horizontally. For \(g(x)\), use \(x = -2, -1, 2,\) and \(7\), which results in \((-2,0), (-1,1), (2,2),\) and \((7,3)\).

The main takeaway here is the horizontal shift to the left by 2 units. Understanding these shifts can help in graphing and analyzing new functions efficiently.

Nonnegative Domain

In square root functions, the domain is limited to nonnegative numbers. This means the expression inside the square root must be zero or positive.

For \(f(x) = \sqrt{x}\), \(x\) must be 0 or greater. For \(g(x) = \sqrt{x+2}\), \(x + 2\) needs to be at least zero. Thus, \(x\) can be \(-2\) or more.

Emphasizing the nonnegative domain is crucial since square roots of negative numbers aren't real. By recognizing this, you can identify valid \(x\)-values and correctly graph the function.

Coordinate System

Plotting functions requires using a coordinate system effectively. The rectangular coordinate system, consisting of the \(x\)-axis and \(y\)-axis, lets us visualize function behavior.

For \(f(x)\) and \(g(x)\), accurately plotting points like \((0,0)\) for \(f\) or \((-2,0)\) for \(g\) is necessary.

Using labeled axes helps in seeing the relationship between functions and their transformations.

• Always ensure your plots are neat and use a consistent scale.
• This allows easy interpretation of the graph's shape and the changes that occur through function transformation.

Function Analysis

Analyzing functions involves not only graphing but also understanding their relations. For \(f\) and \(g\), the analysis focuses on how \(g\) transforms from \(f\).

Once plotted, you can see that \(g(x)\) is shifted 2 units left from \(f(x)\). This indicates a horizontal transformation.

Here are some analysis tips:

• Look for shifts, stretches, or reflections in transformations.
• Compare how function values change with different \(x\)-values.
• Assess if the transformation changes the domain or range.

Analyzing these aspects helps grasp the underlying principles of function transformations and relationships.