Question

Sketch the surface given by the function. $$ g(x, y)=\frac{1}{2} x $$

Short answer

The graph of the function \(g(x, y) = \frac{1}{2}x\) is a plane inclined towards the x-axis at a 45-degree angle.

Step-by-step solution

Understanding the equation

The function \( g(x, y)= \frac{1}{2} x \) tells us that the z coordinate (in 3D space) is directly proportional to x and is independent of y. This means all points with the same x coordinate will have the same height or z coordinate, irrespective of what the y coordinate is. So, the graph of the function will be a plane oriented at an angle to the x-axis.

Plotting the function

The function can be graphed by keeping in mind that for each x value, the z value is half of it and it remains at the same height for every y value. Plot a few points on a graph, like \((1,0,0.5)\), \((2,0,1)\), and \((3,0,1.5)\). You can see that with increase in the x coordinate, the z coordinate also increases, but half as fast. Now remember the graph remains at the same height for a given x, no matter what y is. So extend these points into horizontal lines parallel to y-axis. These lines together create the graph of the function, which is a slanted plane.

Confirming the graph

Look at your graph. For any given x coordinate, all the y values should be on the same level, creating a line parallel to the y-axis. All of these lines together form an inclined plane.

Key concepts

3D Coordinate System

The 3D coordinate system extends the familiar 2D Cartesian plane into a third dimension, allowing us to describe the position of any point in three-dimensional space. This additional dimension is usually represented by the z-axis, which is perpendicular to both the x-axis and y-axis.

In a 3D coordinate system, any point can be located using a set of three coordinates: \( (x, y, z) \). The x-coordinate measures the horizontal distance along the x-axis, the y-coordinate measures the depth along the y-axis, and the z-coordinate measures the vertical height along the z-axis. By combining these three values, you can pinpoint the exact location of a point in 3D space.

When graphing functions like \( g(x, y) = \frac{1}{2} x \), it's crucial to recognize that the z-value output by our function will represent the height above the x-y plane. The input variables x and y determine the position within the plane, and the function's formula gives us the corresponding z-value for each (x, y) pair.

Proportional Relationship

A proportional relationship occurs when two quantities maintain a constant ratio between them. In the context of functions and graphing, if one variable is always some constant multiple of the other, this constant ratio reflects a proportional relationship between them.

In the exercise, we encounter the function \( g(x, y) = \frac{1}{2} x \), which signifies a proportional relationship between x and z. Here, the z-value is always one-half of the x-value—it increases or decreases at half the rate of x. The constant of proportionality is \( \frac{1}{2} \).

This relationship is essential when graphing the function, as it tells us how to scale our z-values in comparison to our x-values, which results in a consistent, predictable pattern on the graph—the incline of the plane. It's this steady change, without any influence from the y-value, that makes the plane's slope constant in the x-z direction.

Inclined Plane

An inclined plane, in geometric terms, is a flat surface that slants at an angle relative to a horizontal surface. The function in our exercise graphs to form such a surface, because z changes with x but is not affected by y.

In graphing \( g(x, y) = \frac{1}{2} x \), the points we plot form horizontal lines that are parallel to the y-axis, which reflects that the z-value does not depend on y. Joining these lines together, we get an inclined plane that rises at a constant angle with respect to the x-axis, created by the proportional relationship between x and z.

Such a plane is a fundamental shape in the understanding of 3D geometry, often seen in physical forms such as ramps, hills or roofs. Recognizing and graphing such planes can help students visualize how 3D objects and their respective functions relate to real-world structures and phenomena.