Problem 56 - College Algebra Essentials

Question

Graph each equation in a rectangular coordinate system. $$f(x)=3$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 3\) is a horizontal line parallel to the x-axis passing through the point (0, 3).

Step 1: Understanding the function

The given function is \(f(x) = 3\). This is a constant function, where the value of y, which is 3 in this case, stays the same no matter what x is. This means the graph will be a horizontal line at y=3.

Step 2: Setting up the graph

First, draw the x and y axes on your graph paper. Since the function is \(f(x) = 3\), the graph will be a horizontal line 3 units over the x-axis. Mark the value 3 on the y-axis.

Step 3: Drawing the graph

Draw a straight horizontal line that passes through the point (0,3) on the y-axis. This line represents the equation \(f(x) = 3\), and it should be parallel to the x-axis.

Key Concepts

Understanding the Rectangular Coordinate System

The rectangular coordinate system is the backbone of graphing in mathematics. It consists of two number lines that intersect at a right angle, creating a grid for plotting points. These two number lines are called axes:

• The horizontal line is known as the x-axis.
• The vertical line is referred to as the y-axis.

The point where the x-axis and y-axis meet is called the origin. It serves as the "center" of the coordinate system with coordinates (0, 0). Each point on the plane is identified by a pair of numbers (x, y).
These numbers indicate the position relative to the origin, with 'x' being the horizontal distance and 'y' the vertical distance. The rectangular coordinate system is essential for visualizing relationships between variables in equations and for graphing functions like constant functions.

Delving into Constant Functions

A constant function is a type of linear function where the output value does not change, regardless of the input. This means that no matter what value you substitute for 'x', the y-value remains constant.
In mathematical terms, a constant function is written as:

• \(f(x) = c\), where 'c' is a constant real number.

So, if you have a function such as \(f(x) = 3\), it means that for any x-value, the output is always 3. This is a useful concept because it simplifies graphing: you don’t need to calculate different points to plot your graph.
Constant functions often occur in real-world scenarios where a quantity remains unchanged regardless of variable conditions. This makes understanding and graphing them straightforward once you grasp their nature.

The Horizontal Line and Graphing Constant Functions

When graphing a constant function like \(f(x) = 3\), you are essentially drawing a horizontal line on the graph. A horizontal line has a consistent y-value, no matter what x-value you choose.
To graph a horizontal line:

• First, find the constant y-value of your function, which in this case is 3.
• Next, locate this y-value on the y-axis. This will be your starting point.
• Then, draw a straight line parallel to the x-axis, ensuring it passes through the point you marked on the y-axis.

The line extends infinitely in both the positive and negative x-directions but remains fixed at y = 3. This makes it parallel to the x-axis and easy to spot on the graph. By graphing in this way, you effectively show that the output of the function is consistently 3, demonstrating the nature of constant functions.