How to Draw Contour Lines of a Multivariable Function

Algebra and Function Practice

A contour line (level curve) shows all points in the domain where a multivariable function equals a chosen constant. Drawing contours by hand builds intuition for gradients, surfaces, and optimization. The method below uses algebra to find curve families, a sampling grid to locate points, and a consistent convention for connecting them. Because hand-drawn curves vary, the rubric rewards correct level values, correct point placement, smooth connections, and clear labeling rather than pixel-perfect shapes.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Set up the level equation and identify the curve family

Given z=f(x,y) and a level value c, write f(x,y)=c. Simplify algebraically to recognize the curve type. For f(x,y)=x^2+y^2, the level x^2+y^2=c is a circle of radius sqrt(c) when c>0, a single point when c=0, and empty when c<0. For f(x,y)=x+y, the level x+y=c is the line y=c-x. For f(x,y)=xy, the level xy=c is y=c/x for x not zero, a hyperbola branch pair.

Choose level values that produce distinct, readable curves. For circles, c=1,4,9 give radii 1,2,3. For lines, c=-2,0,2 give parallel lines. For hyperbolas, c=1 and c=-1 give branches in opposite quadrants. Label each contour with its c value near an end or along the curve.

1. Write f(x,y)=c and simplify.
2. Identify the curve family and any restrictions such as x not zero.
3. Select three to five level values that fit the grid.
4. Compute key intercepts or radii for each level.
5. Sketch axes, scale, and grid before plotting points.

Sample points on a grid and connect them smoothly

Use a table of x values and solve for y, or vice versa, to get points that satisfy f(x,y)=c. For x^2+y^2=4, choose x=-2,-1,0,1,2 and compute y=±sqrt(4-x^2). Plot both signs. For x+y=2, choose x=-1,0,1,2 and compute y=3,2,1,0. For xy=1, choose x=0.5,1,2 and compute y=2,1,0.5; also plot the branch for negative x and y.

Connect points with a smooth curve that respects symmetry and asymptotes. Do not connect across a gap where the function is undefined. Use a light pencil for construction points and a darker line for the final contour. Keep spacing between contours consistent with the level values: equal c steps usually give equal visual spacing for linear functions, but not for nonlinear ones.

1. Make a small table of at least four points per contour.
2. Plot points lightly and check they satisfy f(x,y)=c.
3. Connect points in order without sharp corners unless the true curve has them.
4. Mark asymptotes or excluded values with dashed lines if helpful.
5. Label each contour with its c value.

Use a consistent convention and check your drawing

Adopt a stated convention: for example, solid curves for positive levels, dashed for negative, and a dot for zero if it is a single point. State the convention on the worksheet. Because handwriting and curve shapes vary, a valid alternative is any drawing that passes through the computed points, has the correct curve family, and is labeled correctly.

Check by picking one plotted point per contour and substituting into f(x,y). The result must equal c. Check that contours do not cross unless the function has a saddle where levels meet at a point. Check that labels match the level values and that axes have consistent units.

1. State your line convention at the top of the page.
2. Substitute one point per contour into the formula.
3. Verify no contour crosses another except at a shared point.
4. Confirm labels and scale are readable.
5. Compare spacing to expected behavior near the origin or asymptotes.

Worked example

Draw contour lines for f(x,y)=x^2+y^2 at c=1,4,9 on a grid from -4 to 4 in both axes. Label each contour.

Four practice exercises

Common mistakes

• Plotting points that do not satisfy f(x,y)=c. Substitute each point into the equation before connecting. If it fails, recompute y from the level equation.
• Connecting hyperbola branches across the axes. Keep branches separate and mark excluded values such as x=0 with a dashed line.
• Using equal spacing for nonlinear contours without checking. Compute radii or intercepts for each level; spacing changes with c for circles and hyperbolas.
• Forgetting to label contours with their c values. Write the level value near an end of each curve and include a small legend if needed.

Self-check rubric

• Level equations are written correctly for each chosen c.
• At least four points per contour are plotted and satisfy f(x,y)=c.
• Curves are smooth and match the correct family (circle, line, hyperbola).
• Contours are labeled with their c values and axes have consistent units.
• A stated line convention is followed, and valid alternative curve shapes are accepted if points and labels are correct.

Questions and answers