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z-intercept

z-intercept

A point at which a graph intersects the z-axis.

 

 

See also

x-intercept, y-intercept

Key Formula

z-intercept=(0, 0, z)where z=f(0, 0)\text{z-intercept} = (0,\, 0,\, z) \quad \text{where } z = f(0,\, 0)
Where:
  • xx = First coordinate, set to 0 to find the z-intercept
  • yy = Second coordinate, set to 0 to find the z-intercept
  • zz = The value obtained after substituting x = 0 and y = 0 into the equation

Worked Example

Problem: Find the z-intercept of the plane 2x + 3y + 4z = 12.
Step 1: To find the z-intercept, set x = 0 and y = 0 in the equation.
2(0)+3(0)+4z=122(0) + 3(0) + 4z = 12
Step 2: Simplify the left side. Both terms with x and y vanish.
4z=124z = 12
Step 3: Solve for z by dividing both sides by 4.
z=124=3z = \frac{12}{4} = 3
Step 4: Write the z-intercept as a point in 3D space.
(0, 0, 3)(0,\, 0,\, 3)
Answer: The z-intercept is the point (0, 0, 3).

Another Example

This example uses a nonlinear surface (a paraboloid) rather than a plane, showing that the same method — setting x = 0 and y = 0 — works for any equation in three variables.

Problem: Find the z-intercept of the surface z = x² + y² + 5.
Step 1: Set x = 0 and y = 0 in the equation.
z=(0)2+(0)2+5z = (0)^2 + (0)^2 + 5
Step 2: Simplify to find z.
z=0+0+5=5z = 0 + 0 + 5 = 5
Step 3: State the z-intercept as a point.
(0, 0, 5)(0,\, 0,\, 5)
Answer: The z-intercept is the point (0, 0, 5).

Frequently Asked Questions

How do you find the z-intercept of an equation?
Set both x and y equal to zero, then solve the equation for z. The resulting point has the form (0, 0, z). This works for planes, surfaces, and any other graph in three-dimensional coordinate space.
What is the difference between a z-intercept, x-intercept, and y-intercept?
Each intercept tells you where a graph crosses a particular axis. The x-intercept is where the graph crosses the x-axis (set y = 0 and z = 0). The y-intercept is where it crosses the y-axis (set x = 0 and z = 0). The z-intercept is where it crosses the z-axis (set x = 0 and y = 0). In 2D problems there is no z-axis, so z-intercepts only arise in 3D.
Can a graph have more than one z-intercept?
Yes. If substituting x = 0 and y = 0 yields more than one value of z, the graph has multiple z-intercepts. For example, the sphere x² + y² + z² = 9 gives z = 3 and z = −3, so it has two z-intercepts: (0, 0, 3) and (0, 0, −3). Some graphs may have no z-intercept at all if the equation has no solution when x = 0 and y = 0.

z-intercept vs. y-intercept

z-intercepty-intercept
DefinitionPoint where a graph crosses the z-axisPoint where a graph crosses the y-axis
How to find itSet x = 0 and y = 0, solve for zSet x = 0 (and z = 0 in 3D), solve for y
Coordinate form(0, 0, z)(0, y) in 2D or (0, y, 0) in 3D
Typical context3D graphs: planes, surfaces, space curves2D and 3D graphs: lines, curves, planes

Why It Matters

You encounter z-intercepts in multivariable calculus and 3D analytic geometry when graphing planes, quadric surfaces, and other surfaces. Finding intercepts on all three axes is one of the quickest ways to sketch a plane in three-dimensional space. In physics and engineering, the z-axis often represents height or depth, so the z-intercept can carry direct physical meaning — for instance, the initial altitude of a trajectory when horizontal coordinates are zero.

Common Mistakes

Mistake: Setting only x = 0 (but not y = 0) when looking for the z-intercept.
Correction: The z-axis is the set of all points where both x = 0 and y = 0. You must substitute both x = 0 and y = 0 into the equation to find where the graph meets the z-axis.
Mistake: Confusing the z-intercept with a trace or cross-section in the xz-plane or yz-plane.
Correction: A trace is an entire curve obtained by setting one variable to zero. The z-intercept is a single point (or a few points) found by setting two variables to zero. Make sure you set both x and y to zero, not just one.

Related Terms