Problem 40 - Precalculus Essentials

Question

Graph the given functions, \(f\) and \(g,\) in the same rectangular coordinate system. Select integers for \(x,\) starting with -2 and ending with \(2 .\) Once you have obtained your graphs, describe how the graph of \(g\) is related to the graph of \(f .\) $$f(x)=x, g(x)=x-4$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x - 4\) is a vertical translation of the graph of the function \(f(x) = x\) four units down.

Step 1: Graph the function \(f(x)\)

Start by creating a table of values for the function \(f(x) = x\) for \(x\) from -2 to 2. Plot these points on a rectangular coordinate system. This gives a straight line.

Step 2: Graph the function \(g(x)\)

Next, create a table of values for the function \(g(x) = x - 4\) for \(x\) from -2 to 2. Plot these points on the same rectangular coordinate system. This also gives a straight line.

Step 3: Compare the graphs of the functions

Look at the two graphs created. Describe how the graph of function \(g(x)\) is related to the graph of function \(f(x)\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is the foundation of graphing functions like the ones presented in your exercise. Often referred to as the Cartesian plane, it consists of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin, denoted as the point (0,0). Each point in this system is represented by an ordered pair (x, y), indicating its coordinates along the x-axis and y-axis, respectively.

To graph a function, we use these axes to plot points that correspond to pairs of values satisfying the function's equation. For instance, the point where x is 1 and y is 1 for the function f(x) = x would be (1,1). Successfully plotting several such points and connecting them reveals the visual representation of the function. Recognizing the importance of this system is crucial for graphing as it allows us to visualize and analyze functions effectively.

Table of Values

Creating a table of values is a methodical step toward graphing a function. It involves choosing inputs, calculating corresponding outputs, and then listing these pairs in a tabular form. The exercise prompts you to use integers from -2 to 2 for x-values. For the function f(x) = x, simply use the x-values as y-values as well, as the function's output equals its input.

For g(x) = x - 4, you’ll subtract 4 from each x-value to find the y-values. This approach provides a clear insight into what points to plot on the coordinate system. It's an especially useful technique for visualizing linear functions—and more complex functions—before you even draw them on paper or screen. A proper table of values simplifies the graphing process and serves as a practical guide for both learning and checking your work.

Linear Functions

Regarding linear functions, they are one of the simplest and most fundamental types of functions in precalculus. They create straight lines when graphed and have the general form f(x) = mx + b, where m stands for the slope and b for the y-intercept. In your exercise, f(x) = x can be interpreted as having a slope (m) of 1 and a y-intercept (b) of 0, meaning that it passes through the origin.

Similarly, g(x) = x - 4 also represents a linear function, but with a y-intercept of -4. The slope remains 1, indicating that the steepness of the line is identical to that of f(x). Linear functions are pervasive in real-world problems, being the go-to model for relationships with a constant rate of change.

Comparing Graphs

Comparing the graphs of different functions helps to understand the relationship between them. In this task, by graphing both f(x) = x and g(x) = x - 4 on the same coordinate system, we can visually discern that the latter is simply a vertical shift of the former. Specifically, g(x) is the graph of f(x) translated 4 units downwards.

This tells us the two functions have the same shape and the same slope, which is a characteristic of linear functions with parallel graphs. Through comparison, we can also explore other relationships or transformations, such as reflections, stretches, or compressions, and it is a vital skill for analyzing how changes in function formulas affect their graphical representations.