Slices at Fixed y

What a Fixed-$y$ Slice Shows

A surface $z=f(x,y)$ assigns a height to each point $(x,y)$ in its domain. To see how the surface behaves along a particular vertical plane, choose a constant $b$ and restrict attention to points where $y=b$. The plane $y=b$ is parallel to the $xz$-plane; its intersection with the surface is a curve called a vertical trace at fixed $y$.

Finding this trace is direct: substitute $b$ for $y$ in the formula for the surface. The resulting equation gives $z$ in terms of $x$. To sketch it, regard $x$ as the horizontal coordinate within the plane $y=b$, and $z$ as the vertical coordinate. Keep in mind that the curve lies at the fixed $y$-coordinate $b$ in three-dimensional space.

Definition: Let $f:D\to\mathbb{R}$, where $D$ is a subset of the $xy$-plane. The vertical trace of the graph of $f$ in the plane $y=b$ consists of the points $$ (x,b,f(x,b)) $$ for all $x$ such that $(x,b)\in D$. Equivalently, it is the intersection of the graph with the plane $y=b$.

The allowable $x$-values matter: the expression $f(x,b)$ defines a trace only where the original function is defined. If the domain is all of $\mathbb{R}^2$, every real $x$ is available for every $b$. Otherwise, determine which $x$-values satisfy the domain condition after setting $y=b$.

From the Surface Equation to a Trace

The graph of $f$ is the set of points $(x,y,z)$ satisfying $z=f(x,y)$. In the plane $y=b$, the coordinate $y$ is no longer variable. Therefore the two conditions describing the intersection are $y=b$ and $z=f(x,b)$. Together they specify the trace.

\begin{aligned} z&=f(x,y), \qquad y=b,\\ z&=f(x,b). \end{aligned}

This explains why substitution gives the correct curve: it enforces the equation of the plane while retaining exactly the points that also satisfy the surface equation. For a sketch in the fixed plane, plot the relation $z=f(x,b)$ in its $x,z$ coordinates. When describing a point in the original three-dimensional picture, include its fixed coordinate $y=b$.

A reliable procedure is:

Choose the plane.
Set $y=b$, recording the fixed value clearly.

Substitute and simplify.
Write $z=f(x,b)$, keeping $x$ as the variable and checking the domain.

Identify the curve.
Use its equation to recognize the shape and locate useful points such as a vertex or intercept.

Place it in space.
Sketch the curve in the plane $y=b$, then compare its shape and height with traces in other fixed-$y$ planes.

Worked Example: Parabolas at Different Heights

Find and compare the traces of $z=x^2+2y$ at $y=-1$, $y=0$, and $y=2$. The domain is all of $\mathbb{R}^2$, so each trace allows every real $x$.

For $y=-1$, substitute $-1$ for $y$:

$$z=x^2+2(-1)=x^2-2.$$

This is an upward-opening parabola with vertex $(x,z)=(0,-2)$ in the plane $y=-1$. For example, at $x=1$, $z=1-2=-1$. Checking in the original formula gives $1^2+2(-1)=-1$, as required.

For $y=0$, substitution gives

$$z=x^2+2(0)=x^2.$$

Its vertex is $(0,0)$ in the plane $y=0$. At $x=1$, the trace gives $z=1$, and the original formula gives $1^2+2(0)=1$.

For $y=2$,

$$z=x^2+2(2)=x^2+4.$$

This parabola has vertex $(0,4)$ in the plane $y=2$. At $x=1$, $z=1+4=5$, and the original formula gives $1^2+2(2)=5$.

All three traces have the same parabolic shape and the same $x$-coordinate for their vertex. Their vertex heights are $-2$, $0$, and $4$, respectively. Thus, across these selected planes, the trace moves vertically while retaining its shape. This comparison concerns the traces in the chosen planes; it does not claim that one whole trace is above another for every possible choice of functions.

Worked Example: A Family of Lines

For $z=xy+1$, find the traces at $y=-2$, $y=0$, and $y=2$, and compare their slopes and heights.

At $y=-2$,

$$z=x(-2)+1=-2x+1.$$

This is a line. When $x=0$, it has height $z=1$; when $x=1$, it has height $z=-1$. Substitution in the original formula confirms both values: $0(-2)+1=1$ and $1(-2)+1=-1$.

At $y=0$,

$$z=x(0)+1=1.$$

The trace is the horizontal line $z=1$ within the plane $y=0$. For instance, at $x=1$, the original formula gives $1(0)+1=1$.

At $y=2$,

$$z=x(2)+1=2x+1.$$

This line has height $1$ at $x=0$ and height $3$ at $x=1$. The original formula gives $0(2)+1=1$ and $1(2)+1=3$, verifying the trace values.

Each trace passes through the point with $x=0$, $z=1$, but the slopes in the $xz$-coordinates are $-2$, $0$, and $2$. The traces therefore change direction as the fixed $y$-value changes, even though they share this height at $x=0$. In three dimensions, those points are $(0,-2,1)$, $(0,0,1)$, and $(0,2,1)$, which lie in different planes.

Worked Example: A Shifted Parabola

Find the traces of $z=x^2+2xy+2y^2$ at $y=-2$, $y=0$, and $y=1$. Compare the vertex locations and heights.

At $y=-2$, substitution gives

$$z=x^2+2x(-2)+2(-2)^2=x^2-4x+8.$$

Completing the square, $x^2-4x+8=(x-2)^2+4$. The vertex is therefore at $x=2$, with height $z=4$. Checking the original formula at $(x,y)=(2,-2)$ gives $2^2+2(2)(-2)+2(-2)^2=4-8+8=4$.

At $y=0$,

$$z=x^2+2x(0)+2(0)^2=x^2.$$

The vertex is at $x=0$, with height $z=0$. The original formula at $(0,0)$ gives $0^2+2(0)(0)+2(0)^2=0$.

At $y=1$,

$$z=x^2+2x(1)+2(1)^2=x^2+2x+2=(x+1)^2+1.$$

The vertex is at $x=-1$, with height $z=1$. Substituting $(x,y)=(-1,1)$ in the original formula gives $(-1)^2+2(-1)(1)+2(1)^2=1-2+2=1$.

These traces all open upward, but their vertices differ in both horizontal position and height: $(2,4)$, $(0,0)$, and $(-1,1)$ in their respective $xz$-coordinates. Completing the square makes both kinds of change visible. The full three-dimensional vertex points also include the fixed coordinates: $(2,-2,4)$, $(0,0,0)$, and $(-1,1,1)$.

Why the Fixed Coordinate Must Be Kept

A trace equation such as $z=x^2+4$ describes the shape in $xz$-coordinates, but by itself it does not say which plane contains the curve. In the first example, $z=x^2+4$ occurred at $y=2$. The points on that trace have the form $(x,2,x^2+4)$, not $(x,0,x^2+4)$. Keeping the plane label prevents a correct two-dimensional sketch from being misplaced in space.

The traces can also differ in more than height. A fixed value may change coefficients of $x$, changing the direction of a line, or produce a horizontal shift in a parabola. To compare them, identify the equation after substitution first; then compare familiar features such as shape, vertex, and values at selected $x$-coordinates. Comparing only the substituted constants can miss a change in position or slope.

A Common Mistake: Fixing the Wrong Variable

Suppose the surface is $z=x^2+2xy+2y^2$, and the requested plane is $y=1$. A common error is to set $x=1$, producing $z=1+2y+2y^2$. That equation describes a restriction with $x$ fixed, not the requested plane. It still contains $y$ as a variable.

For the fixed-$y$ trace, substitute $y=1$ and leave $x$ variable:

$$ z=x^2+2x(1)+2(1)^2=x^2+2x+2=(x+1)^2+1. $$

The resulting curve lies in the plane $y=1$, has vertex $(-1,1)$ in $xz$-coordinates, and has three-dimensional vertex $(-1,1,1)$. The check is $ (-1)^2+2(-1)(1)+2(1)^2=1$, matching the stated vertex height.

Takeaway: For a trace at fixed $y=b$, substitute $b$ for $y$, not for $x$. Then record that the resulting curve remains in the plane $y=b$.

Check Your Understanding

For each question, state the fixed plane as well as the equation or feature of the trace.

For $z=x^2-y$, what equation describes the trace at $y=3$, and what is its vertex height?
For $z=xy+2$, find the trace at $y=-1$. What is its height when $x=2$?
For $z=x^2+2xy+2y^2$, find the trace at $y=0$ and state its vertex.
In your own words, why does substituting $y=b$ give the intersection with the plane $y=b$?
What information is lost if you write only a trace equation in $x$ and $z$, without recording the fixed value $b$?

What You'll Learn

What a Fixed-\(y\) Slice Shows

From the Surface Equation to a Trace

Worked Example: Parabolas at Different Heights

Worked Example: Parabolas at Different Heights

Worked Example: A Family of Lines

Worked Example: A Family of Lines

Worked Example: A Shifted Parabola

Worked Example: A Shifted Parabola

Why the Fixed Coordinate Must Be Kept

A Common Mistake: Fixing the Wrong Variable

Check Your Understanding

Slices at Fixed y

What You'll Learn

What a Fixed-\(y\) Slice Shows

From the Surface Equation to a Trace

Worked Example: Parabolas at Different Heights

Worked Example: Parabolas at Different Heights

Worked Example: A Family of Lines

Worked Example: A Family of Lines

Worked Example: A Shifted Parabola

Worked Example: A Shifted Parabola

Why the Fixed Coordinate Must Be Kept

A Common Mistake: Fixing the Wrong Variable

Check Your Understanding

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