A function of two variables takes a point (x, y) and returns a number z. The graph is a surface hovering above the xy-plane.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Where they come from
Temperature depends on latitude and longitude. Cost depends on labour and materials. Two inputs is the ordinary case, not the exotic one.
The domain is a region
For one variable the domain was a set of intervals on a line. Now it is a region in the plane, such as a disc or a half-plane.
What restricts it
The same two rules as before. No dividing by zero, and no even root of a negative number.
Traces
Fixing y at a constant and letting x vary gives an ordinary one-variable curve. Slicing a surface into traces makes it readable.
The shape to know
z = x² − y² curves upward along one axis and downward along the other. That is a saddle, and no single-variable intuition prepares you for it.
Two inputs, one output
z = f(x, y) assigns a height to each point of the plane, so the graph is a surface in space. The domain is a region of the plane rather than an interval.
The domain is a region
Restrictions like a square root or a denominator now carve out areas rather than intervals. Sketching the domain in the plane is usually the first step in any multivariable problem.
Where they arise
Temperature across a plate, elevation across terrain, profit as a function of two prices, concentration across a region. Any quantity depending on two inputs is a function of two variables.
And more variables still
Three or more inputs are common and cannot be graphed at all. The algebra extends unchanged, which is why the theory is developed to work without pictures.
Step 2: Try It Yourself
Tap and try it out.
- The surfacez = a(x² − y²)
- Height at the origin0
A saddle curves upward along one axis and downward along the other. The origin is a critical point that is neither a maximum nor a minimum.
- The surfacez = a(x² + y²)
- Height at the origin0
Each line holds one coordinate fixed and varies the other, so every line is a cross-section of the surface.
Step 3: Watch an Example
One step at a time.
Watch Nadia Evaluate a Surface
Nadia evaluates f(x, y) = x² + 3xy − y at the point (2, 1).
- Step 1
She substitutes x = 2 and y = 1 into each term.
Step 4: Your Turn
Practice makes it stick.
The Plate
Problem 1 of 2
Temperature is T(x, y) = 40 − x² − y² degrees. What is the temperature at (3, 4)?
The Job
Problem 2 of 2
Cost is C(h, m) = 25h + 4m dollars for h hours and m materials. What is the cost at h = 6, m = 10?
Two Inputs at Once
1 of 8
f(x, y) = x + y². What is f(3, 2)?
2 of 8
f(x, y) = xy. What is f(−4, 5)?
3 of 8
f(x, y) = x² − y². What is f(5, 3)?
4 of 8
f(x, y) = x² − y². What is f(3, 5)?
5 of 8
f(x, y) = 1/(x − y). Is (4, 4) in the domain? 1 yes, 0 no.
6 of 8
The trace of z = x² + y² at y = 0. Which curve is it? 1 line, 2 parabola, 3 circle.
7 of 8
Match each surface to its shape.
Tap a card on the left to start.
8 of 8
f(x, y) = 40 − x² − y². What is the largest value it can take?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x, y) = 2x² + y. What is f(3, −4)?
Question 2 of 2
What does the graph of a function of two variables look like?
What You Learned
- A function of two variables sends a point in the plane to a height, so its graph is a surface.
- The domain is now a region of the plane rather than intervals on a line.
- Fixing one variable slices the surface into an ordinary one-variable trace.