Problem 52 - 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 \(f\). $$ \begin{aligned} &f(x)=\sqrt{x} \quad(x=0,1,4,9) \text { and }\\\ &g(x)=\sqrt{x}+2 \quad(x=0,1,4,9) \end{aligned} $$

Step-by-Step Solution

Answer

The graph of g is a vertical shift of 2 units up from the graph of f.

Step 1: Graphing Function \(f\)

Plot the function \(f(x) = \sqrt{x}\) using the given x-values (0,1,4,9). This will give you the ordered pairs (0,0), (1,1), (4,2), and (9,3). Plot these on your graph.

Step 2: Graphing Function \(g\)

Next, plot the function \(g(x) = \sqrt{x} + 2\) using the same x-values as before. This will give us the ordered pairs (0,2), (1,3), (4,4), and (9,5). Plot these points on the same graph as used for \(f\).

Step 3: Comparing the Graphs

After plotting both functions on the same graph, observe that function \(g\) appears to be a vertical shift of function \(f\). Each point on \(g\) is two units above the corresponding point on \(f\), indicating a vertical shift of two units upwards.

Key Concepts

Function Transformations

Function transformations are changes made to the original function that result in a new graph. These changes can affect the position, shape, or orientation of the function's graph on a coordinate plane. A simple way to think about transformations is to consider moving or adjusting the graph in some way. Transformations include shifting, reflecting, stretching, or compressing the graph. Each of these behaves in different ways based on the function and the type of transformation. For example, the function \(g(x) = \sqrt{x} + 2\) is derived from the function \(f(x) = \sqrt{x}\) by applying a vertical shift. In essence, transformations help us visualize how subtle changes in a function's equation modify its graphical representation, making it a powerful tool for analyzing and understanding different functions.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane used for graphing equations. It consists of two perpendicular axes — the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is identified by an ordered pair \((x, y)\), where \(x\) represents the horizontal distance from the origin (intersection of the axes) and \(y\) represents the vertical distance.

This system is crucial for plotting functions like the square root functions \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{x} + 2\). By using this coordinate system, we can clearly visualize how each function behaves across different values of \(x\), allowing us to see patterns, transformations, and intersections in graphing with ease.

Plotting Ordered Pairs

Plotting ordered pairs is a vital part of graphing functions. An ordered pair consists of two values: the first represents the x-coordinate and the second the y-coordinate. These pairs correspond directly to points on the rectangular coordinate system.

For the function \(f(x) = \sqrt{x}\), we are given ordered pairs such as \((0,0), (1,1), (4,2), (9,3)\). To plot these points, find each \(x\) value on the horizontal axis and match it to the corresponding \(y\) value on the vertical axis, marking the point where they intersect. Repeat this process for the function \(g(x) = \sqrt{x} + 2\) with its ordered pairs \((0,2), (1,3), (4,4), (9,5)\), making sure both sets of points appear on the same graph.

• Ensure accuracy by double-checking coordinates before plotting.
• Clearly mark each point to avoid confusion.
• Use these points to help draw the curves of each graph.

Plotting ordered pairs provides the groundwork for drawing and understanding the shape of a function's graph.

Vertical Shifts

Vertical shifts occur when a graph moves up or down from its original position. These shifts don't change the shape of the graph; instead, they move every point of the graph the same distance in the vertical direction. In the context of square root functions, a vertical shift is a common transformation. For example, when transforming \(f(x) = \sqrt{x}\) to \(g(x) = \sqrt{x} + 2\), all the points are shifted two units upward.

This translation is represented by the "+2" in function \(g(x)\). The vertical shift is straightforward because it adds a constant to the output (y-value) of every ordered pair corresponding to the graph of the original function.

• A positive constant, such as "+2", shifts the graph up.
• A negative constant would shift the graph down.

Understanding vertical shifts helps in quickly identifying changes in the function and predicting how the graph will look after being transformed.