Question

Sketch the surface given by the function. $$ f(x, y)=6-2 x-3 y $$

Short answer

Sketching the surface of the given function, we get a plane with z-intercept at (0,0,6), x-intercept at (3,0,0) and y-intercept at (0,2,0). The slope along x axis is -2 and along y axis is -3.

Step-by-step solution

Identify the intercepts

Set each variable to 0 one at a time to find the intercepts. For example, setting x and y to zero, f(0,0) results in the z-intercept at (0,0,6). Likewise, setting y and z to zero results in x-intercept at (3,0,0) and setting x and z to zero results in y-intercept at (0,2,0).

Calculate the slopes

The formula of the function can be written as \(z = 6 - 2x - 3y\). Here, -2 and -3 are the slopes or gradients along x axis and y axis respectively.

Plot the intercepts and slopes

Draw the x, y, z-axis on a 3D space. Plot the intercepts obtained in step 1. Also, the slopes obtained in step 2 would give the direction in which the function is decreasing. You can then sketch the surface with the given decreasing direction from the intercepts.