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Chapter 1: Problem 56

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

Short Answer

Expert verified
The graph of \(f(x)=3\) is a straight line parallel to the x-axis and passing through the point (0,3) on the y-axis.

Step by step solution

01

Understanding the Equation

First, you need to understand the function equation \(f(x)=3\). Here, \(f(x)\) is equivalent to the y-coordinate on the graph. Hence, the equation tells us that for every x-coordinate, the corresponding y-coordinate (y = f(x)) is always 3.
02

Plotting Points

To begin graphing the equation, some representative points could be chosen. For example, the points could be (-2,3), (0,3), (2,3) - these points all have y value as 3. Choose more points for a more accurate graph.
03

Drawing the Graph

After plotting the points on the rectangular coordinate system, a straight horizontal line running through the y-axis at 3 should be drawn. That line would represent the function \(f(x)=3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function
Understanding a constant function is the first step in graphing the equation \(f(x) = 3\). A constant function is a type of function where the output value, which is often a y-coordinate in graphing terms, remains the same regardless of the input value, which is the x-coordinate. This means:
  • No matter what x-value you choose, the y-value will always be the same for all points on the line.
  • In this case, y is always equal to 3, making it a constant value.

The graphical representation of a constant function is a straight line parallel to the x-axis. For the function \(f(x) = 3\), this means every point on this graph will have a y-coordinate of 3, forming a horizontal line.
Graphing Equations
Graphing an equation involves representing its solutions visually on a coordinate plane. When graphing \(f(x) = 3\), we plot points where the y-value is always 3, regardless of the x-value.
  • Start by selecting several x-values, such as -5, 0, and 5.
  • For each x-value, compute the y-value using the function. Here, it is always y = 3.
Thus, the points (-5, 3), (0, 3), and (5, 3) all lie on this line.
Connecting these points will form a horizontal line, demonstrating that the function is consistently returning a y-value of 3. This horizontal line is the graph of \(f(x) = 3\). It is important to remember that each point on this line is a solution to the equation.
Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian plane, is essential for graphing equations like \(f(x) = 3\). It consists of two axes:
  • The horizontal x-axis
  • The vertical y-axis

These axes intersect at a point called the origin (0,0). Each point in this system is defined by an ordered pair \((x, y)\).
In the case of the equation \(f(x) = 3\), the rectangular coordinate system allows us to plot the points visually, showing their consistent y-value of 3 across various x-values.
Utilizing a grid or graph paper can aid greatly in accurately plotting the points and drawing the line, providing a clear visual confirmation of the constant function and the concept of graphing within this system.

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Most popular questions from this chapter

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