Graphing a piecewise function means plotting each rule on its own restricted interval, then marking endpoints as closed or open so the graph passes the vertical line test. You need three things before you draw anything: the domain breakpoints, the endpoint status at each break, and one sample point per interval. Get those right and the picture is mostly arithmetic.
Here's the part most tutorials rush past. A piecewise graph fails not because the algebra is hard, but because an endpoint sits on the wrong interval. Desmos, TI-84, and the College Board's AP Calculus materials all treat a solid dot as inclusion and an open circle as exclusion, and a graph with a mismatched dot produces two outputs at one input β which means it isn't a function at all.
Read the Brackets and Breakpoints Before Touching Graph Paper
Start with notation. Curly braces group the rules; the "if" clause beside each rule is the interval where that rule applies. In f(x) = { x + 1 if x < 2; 5 if x = 2; xΒ² if x > 2 }, the breakpoint is x = 2, and three separate pieces claim territory on either side of it.
Bracket type is the whole game. A strict inequality like x < 2 excludes 2, so that piece gets an open circle at x = 2. An inclusive inequality like x β₯ 2 includes it, so that piece gets a closed dot. The equals-sign piece above is a single isolated point at (2, 5) β a hole-punch of a graph that students routinely miss.
Write the domain of each piece as an interval before graphing. For the example, the intervals are (ββ, 2), exactly {2}, and (2, β). Laying those three lines out on paper forces you to notice gaps, overlaps, and the lone point, which is faster than discovering them mid-plot.
If two rules both claim an input, the function is undefined there as written. Flag it immediately, because that's usually a typo in the problem. Teachers build these problems so every input lands in exactly one interval.
Confirm Which Endpoint Is Closed With One Substitution
Take the breakpoint value and evaluate every rule that borders it. At x = 2 above, the left rule gives 2 + 1 = 3, the right rule gives 2Β² = 4, and the equals piece gives 5. Three different heights at one x-value is fine β as long as only one is actually plotted.
That's the test. The closed dot belongs to the piece whose inequality includes the breakpoint. Here, x = 2 is claimed by the middle rule (x = 2) with the closed point at (2, 5). The left piece ends with an open circle at (2, 3); the right piece begins with an open circle at (2, 4).
Substituting is not optional busywork. A closed dot at the wrong height is the single most common graded error in piecewise problems, and one substitution catches it before you've drawn a single line.
For jump discontinuities β where the left and right heights differ, as they do at 3 versus 4 β the gap between the open circles is the jump. Note its size if the question asks about continuity later.
Build a Sample-Table for Each Interval in Order
One table per interval, two or three points each, ordered left to right. The left piece x + 1 on (ββ, 2) needs points like x = 0 β 1 and x = 1 β 2. The right piece xΒ² on (2, β) needs x = 3 β 9 and x = 4 β 16. The equals piece is already plotted.
Pick values inside the interval, never on the excluded endpoint. Sampling x = 2 in x + 1 gives a point you'll then circle open anyway, which wastes a row and invites the dot error above.
Linear pieces need two points. Quadratic and higher pieces need three so the curve's bend shows. Absolute-value pieces need the vertex plus one point on each arm.
Keep the tables in one column on scratch paper so the pieces stack visually in the same order they'll appear on the axes. This mirrors how the graph reads left to right.
Plot Each Piece on Its Own Interval and Connect
Draw one piece at a time, left to right, and stop at the breakpoint. A constant piece is a horizontal segment. A linear piece is a straight segment with slope from its rule. A quadratic piece is a curve that bends away from its vertex.
Connect the sample points with the correct shape, not with straight lines automatically. The rule tells you the shape: xΒ² curves, |x| makes a V, and 1/x approaches an asymptote. If the interval ends before the curve's natural endpoint, the graph just stops there.
Mark the endpoint status as you finish each piece. Closed dot for included values, open circle for excluded. Draw the open circle slightly hollow so it reads clearly at a glance and survives a photocopier.
Then step back and scan left to right. You should see a continuous segment, a jump, or a hole β and every x-value in the stated domain should have exactly one plotted point.
Mark Closed Dots and Open Circles With a Consistent Convention
Adopt one convention and use it identically on paper, in Desmos, and on a calculator. On paper: a filled circle means included, an unfilled circle means excluded. In Desmos, type the endpoint as a point and toggle the fill style in the point's settings panel β the default is filled, so open endpoints require a manual change.
On a TI-84 Plus CE, the piecewise template lives under math β B: piecewise(. Enter the expression, then the condition, then the next expression and condition. Endpoint dots aren't automatic; you add them with the 2nd β draw β point menu or by plotting points in 2nd β stat plot.
Inconsistent conventions cause silent errors. A filled dot on an excluded endpoint claims an output that doesn't exist. Most graphing software will happily draw the wrong thing if you tell it to.
Label the endpoints with coordinates on graded work. (2, 5) closed versus (2, 3) open tells the grader exactly what you decided, even if the drawing is imperfect.
Check the Vertical Line Test and Domain Coverage
Slide an imaginary vertical line across the finished graph. It must cross the curve exactly once at every x-value in the domain and zero times outside it. Two crossings at any x means an endpoint error or an overlap you missed.
Then check coverage in the other direction. Does every stated interval have ink on it? A piece that's supposed to run to infinity should have an arrow or a clear continuation. A piece that's supposed to stop at x = 2 should visibly stop.
Pay special attention to isolated points. The lone dot at (2, 5) sits above or below the other pieces and is easy to leave out of the final drawing. It's also the point a vertical line at x = 2 must hit.
If a breakpoint has no dot at all and the pieces don't touch, the function is undefined there. That's unusual and worth double-checking against the original problem.
Enter It in Desmos, TI-84, and a Spreadsheet Without Syntax Errors
Each tool wants the piecewise definition in a different shape. Getting the syntax right the first time saves a round of debugging.
| Platform | Syntax approach | Endpoint handling | Common failure |
|---|---|---|---|
| Desmos (web/app) | Type each piece on its own line with a brace condition, e.g. y = x+1 {x < 2} | Add a separate point; toggle fill in point settings | Forgetting the condition, so the line runs full-width |
| TI-84 Plus CE | math β B: piecewise( with expression and condition pairs | Dots added manually via 2nd β draw β point | Leaving the calculator in connected mode, which bridges the jump |
| Google Sheets | =IF(condition, rule1, rule2) in a helper column, then insert a scatter chart | No open-circle rendering; separate a gap with a blank cell | Line chart connecting across the undefined gap |
| GeoGebra | If(condition, expression) command, one per piece | Right-click point β object properties β fill | Mixing If syntax with braces copied from Desmos |
The table hides one real gotcha: the TI-84's default connected graphing mode draws a line between consecutive plotted points even across a jump, so a piecewise function with a discontinuity looks continuous. Switch to dot mode in mode or restrict the window so the gap sits between screen columns.
In Sheets, a line chart will connect a blank cell's neighbors unless you set the chart to "Treat blank cells as gaps." That setting is under the chart editor's setup panel, and it's off by default.
Desmos is the fastest for checking your hand-drawn answer because it renders open endpoints faithfully β but only if you add them as points. The brace condition alone won't draw a circle.
Work Through a Three-Piece Problem With a Jump
Take f(x) = { x + 1 if x < 2; 5 if x = 2; xΒ² if x > 2 } and graph it end to end.
Step 1 β Mark the breakpoints. x = 2 only. The intervals are (ββ, 2), {2}, and (2, β).
Step 2 β Substitute at the break. Left rule: 2 + 1 = 3. Right rule: 2Β² = 4. Equals piece: 5. The closed dot goes to (2, 5) because the middle rule includes x = 2.
Step 3 β Sample the left piece. x = 0 β 1, x = 1 β 2. Draw the line y = x + 1 through those points, stopping with an open circle at (2, 3).
Step 4 β Sample the right piece. x = 3 β 9, x = 4 β 16. Draw the parabola y = xΒ² from (2, 4) open, curving upward.
Step 5 β Plot the isolated point (2, 5) as a filled dot floating above both pieces.
Step 6 β Check the vertical line. At x = 2 it hits exactly one point, (2, 5). At x = 1.9 it hits the line; at x = 2.1 it hits the parabola. The graph passes.
The end result is a line climbing to a gap, a floating dot, then a parabola launching from a slightly higher open circle. The jump from 3 to 4 is invisible except as empty space, while the dot sits above both.
Verify Continuity Endpoint by Endpoint
A piecewise function is continuous at a breakpoint when three things hold: the left-hand limit exists, the right-hand limit exists, and both equal the function's value there. The example fails all three conditions at x = 2, so it's discontinuous β a jump plus a relocated point.
For the left limit, evaluate the left rule as x approaches 2: x + 1 β 3. For the right, xΒ² β 4. Since 3 β 4, no amount of endpoint fiddling makes it continuous.
Continuity matters because it changes what the graph looks like. A continuous piecewise function has no gaps and no floating dots β the pieces meet. Students often assume all piecewise functions are discontinuous, which is false; f(x) = { x if x < 0; xΒ² if x β₯ 0 } is perfectly continuous because both pieces hit 0 at x = 0.
Check continuity before submitting any graph. The vertical line test confirms it's a function; the endpoint check confirms whether it's continuous. Those are different questions.
Frequently Asked Questions
Can two pieces of a piecewise function overlap at the same x-value?
Not in a valid function definition. If two rules both include a breakpoint, that input produces two outputs, violating the vertical line test. When it happens, the problem is miswritten or you've misread a bracket β check the strict versus inclusive inequality on each side before graphing.
What does an open circle mean on a piecewise graph?
It means the function does not take that value at that x. The point is excluded, usually because the interval uses a strict inequality there. The actual value at that x comes from whichever piece has the closed dot, or the function is undefined there if no piece includes it.
Why does my TI-84 draw a line across the gap?
The calculator's default graphing mode is connected, which draws segments between consecutive plotted points regardless of discontinuities. Switch to dot mode in the mode menu, or set a window where the gap falls between screen columns, and the jump will appear as it should.
Is every piecewise function discontinuous?
No. A piecewise function is continuous wherever its pieces meet at the breakpoints. Absolute value, f(x) = |x|, is piecewise and continuous everywhere. Continuity depends on whether the left and right limits agree with the function's value, not on whether the definition is split.
How many points do I need to graph each piece?
Two points for a linear piece, three for a quadratic or absolute-value piece, one for a constant piece, plus endpoint dots. Fewer points risks a wrong shape; more wastes time. The endpoint substitution always adds one or two more markers regardless of piece type.
How do I graph a piecewise function in Desmos without the line running forever?
Add a brace condition after the expression, like y = x + 1 {x < 2}, so Desmos restricts the domain. The condition is the piecewise definition β without it, Desmos draws the full line. Add endpoint points separately and toggle their fill in the point settings.
What's the difference between a jump discontinuity and a removable one?
A jump means the left and right limits exist but differ, as in the 3-versus-4 example. A removable discontinuity means the limits agree but the function's value is elsewhere or missing β like a hole with a misplaced dot. Both are graphed with open circles, but only the removable case could be fixed by redefining one point.
Do colleges test piecewise graphing on the AP Calculus exam?
Yes, piecewise functions appear on AP Calculus AB and BC in limits, continuity, and differentiability questions. The College Board's course description lists piecewise-defined functions among the function types students must handle, and free-response questions have asked students to determine continuity at breakpoints β the same endpoint check described above.