Problem 21 - Calculus Early Transcendentals

Question

Sketch the graph of the function. $$f(x, y)=3$$

Step-by-Step Solution

Answer

The graph is a horizontal plane at \(z = 3\) parallel to the \(xy\)-plane.

Step 1: Identifying the Type of Surface

The function given is a constant function, specifically, a function of two variables where the output is always 3, regardless of the values of \(x\) and \(y\). This implies that the graph of the function is a surface where the value of \(z\) is always 3.

Step 2: Determining the Dimensions and Shape

Since \(f(x, y) = 3\) does not depend on \(x\) or \(y\), the graph is a horizontal plane in three-dimensional space. This plane is parallel to the \(xy\)-plane and located 3 units above it.

Step 3: Sketching the Plane

To sketch this graph, start by drawing the \(xy\)-plane in 3D space. Then draw a parallel plane above it at \(z = 3\). It should cover the entire \(x\) and \(y\) axes, as the plane extends infinitely in both directions. Label the plane as \(z = 3\).

Step 4: Visualizing the Appearance

Imagine it as a sheet floating parallel to the \(xy\)-plane. The graph is a simple horizontal sheet or plane above the \(xy\)-plane, where any point \((x, y, z)\) on this plane satisfies \(z = 3\).

Key Concepts

Graph of a Constant Function

A constant function is a special kind of function where the output remains the same regardless of the input values. In the case of our function, \(f(x, y) = 3\), this means that no matter what \(x\) or \(y\) you choose, the result is consistently 3.

This tells us that the output or the function's value doesn't rely on the input variables \(x\) and \(y\). It creates a flat surface in space. In three-dimensional graphs, this flatness translates into a very specific geometric shape: a plane. This is because every point on this graph will have its \(z\)-coordinate fixed at 3, no matter the \(x\) and \(y\) coordinates. It is uniform and very straightforward to understand when visualizing functions in a 3D space.

Horizontal Plane

In three-dimensional graphing, a horizontal plane is a flat, two-dimensional surface that stretches infinitely across a 3D coordinate system. For our function, \(f(x, y) = 3\), the graph lies on a horizontal plane.

Since the equation \(z = 3\) represents all points that have a \(z\)-coordinate of 3, this plane is parallel to the \(xy\)-plane. Here's how to visualize it:

• This plane runs parallel to the \(xy\)-plane.
• It is situated 3 units above it within the coordinate system.
• It extends infinitely along both \(x\) and \(y\) directions.

This concept is straightforward because in three-dimensional graphing, planes like these simply represent consistent elevation or depth across a surface.

3D Coordinate System

Understanding the 3D coordinate system is crucial when dealing with graphs like our constant function. This system allows us to plot points in three-dimensional space using three coordinates: \(x\), \(y\), and \(z\). Each point in this space is identified by a triplet \((x, y, z)\).

Here's how it works:

• \(x\) and \(y\) are the horizontal coordinates – they define the direction along the plane's breadth and length.
• \(z\) is the vertical coordinate – it determines the height from the plane's base plane, here specifically at a height of 3.

In our graph, each \((x, y)\) pair relates to a specific point on the \(xy\)-plane while the \(z\) value remains constant at 3. This highlights an important feature of the 3D coordinate system: it allows us to place flat surfaces at different heights with ease.

Visualizing Mathematical Graphs

Visualizing graphs in three-dimensional space can sometimes be daunting, but breaking it down, as with our constant function, makes it easier. Start by understanding the function or equation, like \(f(x, y) = 3\), and recognize what it represents.

For this function:

• Recognize that it's a plane, specifically a horizontal plane at \(z = 3\).
• Look at the relation between \(x\), \(y\), and \(z\) to see if there are consistent outputs.
• Use your imagination to place a sheet or plane parallel to the \(xy\)-plane at this defined height.

Using sketching apps or graphing calculators can provide a visual aid to help grasp these concepts more solidly. By cementing these fundamentals, more complex visualizations become much easier to work with.