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Chapter 15: Problem 117

Complete the following: The graph of a linear function of two variables is a ____ .

Short Answer

Expert verified
The graph of a linear function of two variables is a \(plane\).

Step by step solution

01

Recall basic properties of Linear Functions

A linear function of two variables (generally written as f(x, y) = ax + by + c, where a, b, and c are constants) has a constant rate of change. The graph of a linear function is a straight line in two-dimensional space. When we extend this concept to three dimensions, we need to find the term that describes the shape of the graph in 3D space.
02

Identify the shape of the linear function

In two-dimensional space, the graph of a linear function is a straight line. In three-dimensional space, the graph of a linear function of two variables is a flat surface that consistently maintains the same angle with respect to all axes in the coordinate system. This flat surface is called a "plane." So, the graph of a linear function of two variables is a plane. Therefore, the answer is: The graph of a linear function of two variables is a \(plane\).

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Key Concepts

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

Planes in Three-Dimensional Space
In three-dimensional space, a plane is a flat surface that extends infinitely in all directions. It is like a large piece of paper that has no edges and goes on forever. A plane can be described mathematically using linear equations because they represent the simplest types of surfaces. The equation for a plane can be written as \( z = ax + by + c \), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

Planes have some important properties:
  • They are flat and have no curvature.
  • They can intersect each other at a line.
  • A line perpendicular to a plane is called a normal line.
These characteristics make planes fundamental in geometry and algebra, helping us visualize relationships in 3D spaces.
Constant Rate of Change
The concept of a constant rate of change is crucial in understanding linear functions. In a linear equation, the change in the dependent variable is consistent whenever the independent variable changes. For example, in the equation \(y = mx + c\), \(m\) represents the slope or the rate of change.

This means:
  • For every unit increase in \(x\), \(y\) increases by \(m\).
  • The slope \(m\) is constant, making the rate of change uniform.
The significance of a constant rate of change lies in its predictability, making it easier to forecast future values in a linear relationship.
Two-Dimensional Graphs
Graphs are visual representations of equations and are essential for analyzing linear functions. In two-dimensional space, linear functions appear as straight lines, which can be described by equations like \(y = mx + c\). This is the simplest form of a graph showing a direct relationship between two variables.

Key aspects of two-dimensional graphs:
  • The slope \(m\) determines the line's steepness, influencing how sharply it rises or falls.
  • The intercept \(c\) shows where the line crosses the y-axis.
  • These graphs help identify trends and make predictions based on the linear equation.
Understanding how to read and create these graphs is a fundamental skill in mathematics, helping bridge the gap between numeric expressions and visual interpretation.

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Most popular questions from this chapter

Compute the integrals. HINT [See Example 1.] $$ \int_{1}^{2} \int_{1-2 x}^{x^{2}} \frac{x+1}{(2 x+y)^{3}} d y d x $$

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Under what circumstances would the method of Lagrange multipliers not apply?

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