There are some interesting special cases in which the graphs turn out to be parallel, perpendicular, or coincident with the coordinate planes or axes. Sketch a graph of each of the following equations by drawing their traces as in Problem 1. Then tell what the graph is parallel to, or coincident with.
b. y+z=4
I got y=4 and z=4 and what do i do next? I don't really understand what the question is asking.
c. x+y=0
e. x=5
Any line in the yz plane becomes a plane perpendicular to the yz plane, in xyz space.
Think of the point y=3. When propagated parallel to the x-axis (perpendicular to the y-axis), it becomes a line. Same for a line propagated through space.
x+y=0 is the same thing, parallel to the z-axis.
x=5 is a line (xy) or a plane (xyz) perpendicular to the x-axis.
b. To sketch the graph of the equation y+z=4, we can start by assigning values to one variable and then solving for the other variable.
Let's assign some values to z. Choose a few values for z, such as z=0, z=2, and z=4.
- When z=0, the equation becomes y+0=4, which gives us y=4. So one point on the graph is (0, 4).
- When z=2, the equation becomes y+2=4, which gives us y=2. So another point on the graph is (2, 2).
- When z=4, the equation becomes y+4=4, which gives us y=0. So another point on the graph is (4, 0).
Plotting these points on a 3D coordinate system, you will notice that they form a straight line. This line is parallel to the x-axis because the value of x doesn't affect the equation.
c. To sketch the graph of the equation x+y=0, we can assign values to one variable and solve for the other variable.
Let's assign some values to x. Choose a few values for x, such as x=0, x=2, and x=-4.
- When x=0, the equation becomes 0+y=0, which gives us y=0. So one point on the graph is (0, 0).
- When x=2, the equation becomes 2+y=0, which gives us y=-2. So another point on the graph is (2, -2).
- When x=-4, the equation becomes -4+y=0, which gives us y=4. So another point on the graph is (-4, 4).
Plotting these points on a 2D coordinate system, you will notice that they form a straight line. This line is perpendicular to the x-axis because as the value of x increases or decreases, y remains constant. Therefore, the graph is parallel to the y-axis.
e. To sketch the graph of the equation x=5, we can assign the value of x and observe the value of y and z.
Since x is always equal to 5 for any value of y and z, the graph will be a vertical line parallel to the yz-plane at x=5. It means that as long as y and z are within any range, the value of x will always be 5.
Graphically, it will be a vertical line intersecting the yz-plane at x=5. This line is parallel to the yz-plane.