z-intercept

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Definition

In graphing, it is a point where the graph crosses the z-axis on a three dimensional coordinate system in a Cartesian grid.To find the z-intercept you will need to figure out where the x and y coordinates are equal to 0.

Worked examples

\(2x + 3y + 4z = 12\) → set \(x = 0, y = 0\) → \(4z = 12\) → \(z = 3\) → z-intercept is \((0, 0, 3)\)

To find the z-intercept, set both x and y to zero and solve for z.

\(x - 2y + z = 5\) → \(0 - 0 + z = 5\) → z-intercept is \((0, 0, 5)\)

The z-intercept is where the plane crosses the z-axis.

Common mistakes

Setting \(z = 0\) to find the z-intercept → Set \(x = 0\) and \(y = 0\), then solve for \(z\) The z-intercept is on the z-axis, where x and y are both zero, not where z is zero.
Writing the z-intercept as just the number \(z = 3\) → Write it as the point \((0, 0, 3)\) An intercept is a point with all three coordinates, not just a single value.
\(3x + 2y + 0z = 6\) has z-intercept \((0, 0, 0)\) → This plane has no z-intercept (parallel to z-axis) If the z-term vanishes, the plane is parallel to the z-axis and never crosses it.

Where you'll use it next

Z-intercepts appear when graphing planes and surfaces in multivariable calculus, 3D modeling, and physics (motion in space). You'll use them to sketch graphs and solve systems of equations in three dimensions.

Found in 1 StudyPug lesson

Mastering 3D Coordinate Systems: Navigate the Third Dimension

Multivariable Calculus

Unlock the power of 3D coordinate systems! Learn to visualize points, planes, and shapes in three dimensions. Discover real-world applications in physics, engineering, and computer graphics.

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