Problem 50 - College Algebra 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 \(f\). $$ f(x)=-1, g(x)=4 $$

Step-by-Step Solution

Answer

The graph of function \(f(x) = -1\) is a horizontal line at y = -1. The graph of function \(g(x) = 4\) is a horizontal line at y = 4. Therefore, the graph of g is parallel to the graph of f, and is situated 5 units above it on the y-axis.

Step 1: Identifying the Functions

The functions to graph are \(f(x) = -1\) and \(g(x) = 4\). Since these functions do not depend on \(x\), they will be horizontal lines on the y-axis at y = -1 and y = 4 respectively.

Step 2: Graphing the Functions

Next, graph these functions on a coordinate system. Make the x-axis range from -3 to 3, and y-axis from -2 to 5 for ample space. Draw a horizontal line y = -1 to represent \(f(x)\), followed by another horizontal line at y = 4 to represent \(g(x)\). Mark the points from \(x = -2\) to \(x = 2\) on both lines.

Step 3: Analysing the Relationship

Finally, describe how the graph of g is related to the graph of f. Both graphs are parallel to the x-axis, indicating that both functions are constants. The graph of \(g(x)\) lies above the graph of \(f(x)\) with a vertical distance of 5 units between them, because g(x) = 4 and f(x) = -1, and their difference is 5.

Key Concepts

Constant Functions

A constant function is a simple yet important concept in mathematics. In a constant function, the output value is always the same, regardless of the input. Mathematically, it is represented as \( f(x) = c \), where \( c \) is a constant. This means no matter what \( x \) is, the function will always output the constant value \( c \). For example, if \( f(x) = -1 \), the output will always be -1 for any value of \( x \).

Some key characteristics of constant functions include:

• The graph of a constant function is a horizontal line on the coordinate plane.
• The function does not increase or decrease as \( x \) changes, indicating that the rate of change is zero.
• Constant functions are defined for all real numbers, since they don't depend on the value of \( x \).

Understanding constant functions is essential, especially when analyzing different function graphs because they establish a baseline or reference point for examining other types of functions.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane used for graphing and solving mathematical problems. It consists of two perpendicular axes: the x-axis and the y-axis. These axes intersect at a point called the origin, denoted as \((0, 0)\).

In this system:

• Every point on the plane is represented by an ordered pair \((x, y)\).
• The x-coordinate specifies the horizontal position, while the y-coordinate specifies the vertical position of the point.
• The plane is divided into four quadrants, each defined by the positive and negative directions of the axes.

When graphing functions, like the constant functions \(f(x) = -1\) and \(g(x) = 4\), the rectangular coordinate system helps visualize how these functions behave. In this case, both functions produce straight horizontal lines at their respective y-values (-1 and 4), clearly demonstrating the consistent nature of constant functions.

Parallel Lines

Parallel lines are lines in a plane that never intersect or meet, no matter how far they are extended in either direction. They maintain a constant distance apart and have the same slope. When dealing with functions, the concept of parallel lines often comes into play, especially when comparing similar function types.

In the context of the graphing task, both \(f(x) = -1\) and \(g(x) = 4\) form parallel lines because they are horizontal and never touch or cross each other.

• These lines are parallel to the x-axis, as their slope is zero.
• The vertical distance between them can be thought of as the difference between their y-values, which is 5 units in this case (from -1 to 4).
• This separation reinforces their nature as parallel yet distinct entities within the graph.

Recognizing parallel lines within a graph is crucial for understanding the relationships between multiple functions and can be a useful tool for solving more complex mathematical problems.