Three-Dimensional Coordinates and the Plane Equation
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 4.1: “Functions of Several Variables” (graphs as surfaces) |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/4-1-functions-of-several-variables |
| Supplementary | Stewart Calculus (9th ed.), Section 14.1: “Functions of Several Variables,” pp. 975-976 (Example 5, the linear function as a plane) |
OpenStax Calculus Volume 3 is free and openly licensed. Both texts read the graph of a two-variable function as a surface sitting above its domain in space, the picture to carry forward.
Key idea
You already plot points and read a line as a graph. In one variable, the input $x$ runs along a number line and the output $y$ rises above it, so the graph $y = f(x)$ is a curve in a flat plane. Multivariable calculus adds exactly one direction. The input is now a pair $(x, y)$ that lives on a flat floor, and the output rises straight up out of that floor as a height $z$. The graph is no longer a curve in a plane; it is a surface floating above the floor.
Picture a room. The floor is the $xy$-plane, where your two inputs live. The height off the floor is $z$, the output. A point in the room needs three numbers, $(x, y, z)$: how far along, how far across, how high up. The simplest surface, the one you must be fluent with before everything else, is a flat tilted sheet, a plane, which is what you get when the output depends on the inputs in a straight-line way. A plane in space is the direct cousin of a line in the plane, and its equation $ax + by + cz = d$ is the direct cousin of $y = mx + b$.
So nothing about “input and output” changes. You simply gain one new axis pointing up, and the graph gains one dimension.
Prerequisite Check
Before this lesson, make sure you can do all of the following:
If lines or intercepts are shaky, review Linear Equations and Graphing Functions.
Quick Reference
A point in space is an ordered triple $(x, y, z)$: position along the $x$-axis, along the $y$-axis, and along the $z$-axis (the new vertical direction).
Domain and range, carried up one dimension.
| Single variable | Two variables | |
|---|---|---|
| Input | a number $x$ | a pair $(x, y)$ on the floor |
| Output | a number $y = f(x)$ | a height $z = f(x, y)$ |
| Domain | the set of valid $x$ | the set of valid pairs $(x, y)$, a region in the plane |
| Range | the set of output $y$-values | the set of output $z$-values |
| Graph | a curve in a plane | a surface in space |
The plane equation. \[ ax + by + cz = d \qquad (\text{the space analogue of the line } y = mx + b). \] A function of the form $f(x, y) = px + qy + r$ has graph $z = px + qy + r$, which rearranges to the plane $-px - qy + z = r$.
Intercepts of a plane. Set two of the three variables to zero and solve for the third. The $x$-intercept comes from $y = z = 0$, and so on. Three intercepts are enough to picture the plane in the first part of the room.
The procedure for graphing a plane (flashcard size).
- Write the function as $z = px + qy + r$.
- Find the $x$-intercept ($y = z = 0$), the $y$-intercept ($x = z = 0$), and the $z$-intercept ($x = y = 0$).
- Mark the three intercept points on the three axes.
- The plane through those three points is the graph.
Key Concepts
1. Three numbers locate a point in space
A point in the plane needs two numbers. A point in space needs three, because there is a third independent direction to specify. The triple $(2, 3, 6)$ means: go $2$ along the $x$-axis, $3$ along the $y$-axis, and $6$ up the $z$-axis. The floor of the room is the set of points with $z = 0$, which is the familiar $xy$-plane.
Inline self-check. A point sits directly above the floor point $(2, 3)$ at height $6$. What are its three coordinates, and what is its $z$-coordinate if it sits on the floor instead?
Show answer
Above the floor at height $6$: the point is $(2, 3, 6)$. On the floor: the height is $z = 0$, so the point is $(2, 3, 0)$. Points on the floor are exactly the points with $z = 0$.
2. Domain and range carry up unchanged in meaning
The words “domain” and “range” mean the same thing they always did. The domain is the set of inputs the function accepts; the range is the set of outputs it produces. The only change is the dimension of the input.
For $f(x) = \sqrt{x}$, the domain is $x \ge 0$ (a ray on the number line) and the range is $y \ge 0$. For $g(x, y) = \sqrt{9 - x^2 - y^2}$, the domain is the disk $x^2 + y^2 \le 9$ (a region on the floor) and the range is $0 \le z \le 3$ (the heights the surface reaches). The idea did not change; the input went from a point on a line to a point on a floor.
Recap. If you can say what the domain and range of a single-variable function are, you can say what they are for a two-variable function. Read the domain off the floor and the range off the heights.
3. The graph of $z = f(x, y)$ is a surface above the floor
Definition. If $f$ is a function of two variables with domain $D$, the graph of $f$ is the set of all points $(x, y, z)$ in space with $z = f(x, y)$ and $(x, y)$ in $D$.
You can picture the graph as a sheet lying directly above (or below) the domain $D$ on the floor. Above each allowed floor point $(x, y)$, the sheet sits at height $f(x, y)$. (Source: Stewart 14.1, Graphs, pp. 975, including Figure 5.)
4. A linear function graphs as a plane
The simplest and most important surface is the plane, which is the graph of a linear function.
Example 1. Sketch the graph of $f(x, y) = 6 - 3x - 2y$.
Goal. Recognize the graph as a plane and locate it using intercepts.
Step 1. Write the graph equation: $z = 6 - 3x - 2y$, which rearranges to $3x + 2y + z = 6$. This has the plane form $ax + by + cz = d$.
Step 2. Find the intercepts.
- $x$-intercept ($y = z = 0$): $3x = 6$, so $x = 2$. Point $(2, 0, 0)$.
- $y$-intercept ($x = z = 0$): $2y = 6$, so $y = 3$. Point $(0, 3, 0)$.
- $z$-intercept ($x = y = 0$): $z = 6$. Point $(0, 0, 6)$.
Step 3. The plane passes through $(2, 0, 0)$, $(0, 3, 0)$, and $(0, 0, 6)$. The triangle joining these three intercepts shows the part of the plane in the first part of the room.
\[ \boxed{\text{The graph is the plane } 3x + 2y + z = 6, \text{ through the intercepts } (2,0,0), (0,3,0), (0,0,6).} \]
Recap. A function whose formula is a straight-line combination of $x$ and $y$ always graphs as a plane, and three intercepts pin it down. (A common error is to find only one intercept and try to guess the tilt; three intercepts remove the guessing.) (Source: Stewart 14.1, Example 5, pp. 975-976.)
5. The plane equation is the line equation with one more variable
A line in the plane is $y = mx + b$, or in general form $Ax + By + C = 0$. A plane in space is $z = px + qy + r$, or in general form $ax + by + cz = d$. The grammar is identical; there is just one more variable carrying one more direction. This is why linear functions of two variables play the same central role in multivariable calculus that lines play in single-variable calculus: they are the simplest model, and more complicated surfaces are understood by comparing them to planes locally. That local comparison is the tangent plane, which is the first major payoff of this whole setup in Chapter 14, Section 4.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Using two coordinates for a point in space | writing the apex as $(0, 6)$ | A point in space needs three coordinates: $(0, 0, 6)$ |
| Confusing the floor plane with the graph | calling the domain disk “the graph” | The domain lives on the floor; the graph is the surface above it |
| Finding only one intercept of a plane | marking just $(0, 0, 6)$ | A plane needs three intercepts to picture its tilt |
| Forgetting to rearrange into plane form | leaving $z = 6 - 3x - 2y$ unsorted | Rearrange to $3x + 2y + z = 6$ to read it as $ax + by + cz = d$ |
| Treating domain and range as new ideas | re-deriving what “range” means | They mean exactly what they meant for $f(x)$; only the input dimension changed |
Leveled Practice
Work each problem fully before revealing the answer.
Level 1 -- Direct Application
Problem 1. Give the coordinates of the point that is $4$ along the $x$-axis, $1$ along the $y$-axis, and $5$ up.
Show answer
$(4, 1, 5)$. The three numbers are read in the order $(x, y, z)$.
Problem 2. State the domain and range of $f(x, y) = \sqrt{x} + \sqrt{y}$, drawing on the single-variable idea.
Show answer
Each square root needs a nonnegative inside, so $x \ge 0$ and $y \ge 0$. The domain is the closed first quadrant $\{(x, y) \mid x \ge 0, \; y \ge 0\}$.
Both terms are nonnegative and can be made as large as desired, so the range is $z \ge 0$, that is, $[0, \infty)$.
Problem 3. Find the three intercepts of the plane $2x + 4y + z = 8$.
Show answer
$x$-intercept ($y = z = 0$): $2x = 8$, so $x = 4$. Point $(4, 0, 0)$.
$y$-intercept ($x = z = 0$): $4y = 8$, so $y = 2$. Point $(0, 2, 0)$.
$z$-intercept ($x = y = 0$): $z = 8$. Point $(0, 0, 8)$.
Level 2 -- Graphing Planes
Problem 4. Sketch the graph of $f(x, y) = 12 - 4x - 3y$ by intercepts.
Show answer
Graph equation: $z = 12 - 4x - 3y$, that is, $4x + 3y + z = 12$.
$x$-intercept: $4x = 12$, $x = 3$; point $(3, 0, 0)$.
$y$-intercept: $3y = 12$, $y = 4$; point $(0, 4, 0)$.
$z$-intercept: $z = 12$; point $(0, 0, 12)$.
The graph is the plane through $(3, 0, 0)$, $(0, 4, 0)$, and $(0, 0, 12)$.
Problem 5. Write the plane through the intercepts $x = 2$, $y = 2$, $z = 2$ in the form $ax + by + cz = d$.
Show answer
The intercepts are $(2, 0, 0)$, $(0, 2, 0)$, $(0, 0, 2)$. A plane with $x$-, $y$-, and $z$-intercepts all equal to $2$ is \[ x + y + z = 2. \]
Check: setting $y = z = 0$ gives $x = 2$, and similarly for the others.
Level 3 -- Range from Geometry and Interpretation
Problem 6. The function $g(x, y) = \sqrt{9 - x^2 - y^2}$ has as its graph the top half of a sphere of radius $3$. Using that picture, state the domain and the range.
Show answer
Domain: the square root needs $9 - x^2 - y^2 \ge 0$, that is, $x^2 + y^2 \le 9$, the closed disk of radius $3$ on the floor.
Range: the surface is the upper hemisphere of radius $3$. The lowest height is $0$ (around the rim, where $x^2 + y^2 = 9$) and the highest is $3$ (at the top, above the origin). So the range is $[0, 3]$.
The graph is a surface above the floor, the domain is the disk it sits over, and the range is the set of heights it reaches. This is the full picture the rest of Chapter 14 builds on.
Mastery Checklist
You have mastered this skill when you can, without notes:
Mental Model
Think of a two-variable function as a landscape over a map.
The map is the floor, the $xy$-plane, where your two inputs live. Above each point on the map, the land rises to some height: that height is the output $z = f(x, y)$. The whole landscape, the rolling surface above the map, is the graph. The domain is the part of the map the landscape covers; the range is the set of elevations the land reaches, from its lowest valley to its highest peak.
The flattest possible landscape is a tilted tabletop, a plane. It is the graph of a linear function, and you can place it exactly by marking where it meets each of the three axes. Every more complicated landscape, a dome, a bowl, a saddle, is understood by laying a flat tabletop against it at a point and asking how the land curves away from that flat piece. That flat piece is the tangent plane, and building it is the reason this floor-and-height picture matters.
Connections
Built from
- Linear Equations: The plane equation is the line equation with a third variable, and finding intercepts is the same algebra.
- Graphing Functions: Reading a graph as the picture of input-to-output carries directly from one variable to two.
- Basic Geometric Formulas: Recognizing intercepts and basic shapes supports sketching the surface.
Leads to
- Graphs of Functions: The surface-above-the-floor picture here gets filled in with curved surfaces in the graphs node.
- Functions of Two Variables: The domain-on-the-floor and output-as-height framing is the foundation of the definition of a two-variable function.
Where this shows up
This floor-and-height picture is the stage on which all of multivariable calculus is performed. A partial derivative measures the slope of the landscape in one compass direction; a double integral measures the volume between the landscape and the floor; the tangent plane is the flat tabletop laid against the surface at a point. Beyond mathematics, a function of two variables is how a heat map, an elevation model in geographic software, or a loss surface in machine learning is described: two inputs on a plane and one value rising above each point. Fluency with the three-coordinate picture is what lets all of those be read as surfaces.
Audience Notes
For students who find math intimidating: You are not learning a new idea about functions. You already know input goes in and output comes out. The only new thing is that the input is now a spot on a floor instead of a spot on a line, so the picture gains one upward direction.
For students who want depth: The plane is the unique surface with zero curvature, and the central program of differential calculus in any number of dimensions is to approximate a curved object by a flat one near a point. The tangent plane in Section 14.4 is the two-variable version of the tangent line; the same idea generalizes to tangent hyperplanes in higher dimensions and underlies the derivative as a linear map.
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