Free original lessons
Learn to plot function segments over given intervals and mark open or closed endpoints correctly using simple coordinate arithmetic.
When a function is defined only on a restricted domain, its graph is a segment or ray rather than a full curve. The endpoints of that segment must show whether the endpoint value is included (closed dot) or excluded (open dot). This lesson teaches a reliable procedure: evaluate the function at the interval endpoints, plot intermediate points, connect them smoothly, and mark each endpoint according to the interval notation. All numerical examples are hypothetical and chosen for easy checking.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A domain interval such as [1, 4] means x can equal 1 and 4, so both endpoints are closed. The interval (1, 4) excludes both endpoints, so both are open. Mixed intervals like [1, 4) include 1 but exclude 4. Always translate brackets and parentheses into dot styles before plotting.
The function rule itself does not change; only the portion of the graph within the stated x-values is drawn. Values outside the interval are not plotted, even if the formula could produce them.
Plot the endpoint coordinates and interior points on a coordinate grid. Connect them with a smooth curve or straight line that stays only between the endpoint x-values. Do not extend the graph beyond the interval.
For a linear function the segment is a straight line segment. For a quadratic or other curve, plot enough interior points to show the shape accurately. Label each endpoint with its coordinate pair if space allows.
Use a filled circle for a closed endpoint and an unfilled circle for an open endpoint. This convention is standard in algebra instruction, but handwriting and drawing styles vary. A small filled disk, a solid dot, or a clearly shaded circle all count as closed; an empty ring or circle with visible white center counts as open.
When grading or self-checking, the key requirement is that the inclusion or exclusion is visually unambiguous. A dot that could be mistaken for either style should be redrawn.
A correct graph satisfies three checks: the endpoint x-values match the interval, the endpoint y-values match the function evaluation, and the dot styles match the brackets and parentheses. Interior points should also satisfy the function rule.
A quick arithmetic check at one interior x-value catches most plotting errors. If the plotted point does not match the computed y-value, re-evaluate before connecting.
Graph f(x) = 2x + 1 for the domain [1, 4). Mark endpoints correctly.
Endpoint at x = 1: f(1) = 3, closed dot at (1, 3). Endpoint at x = 4: f(4) = 9, open dot at (4, 9). Interior point at x = 2: f(2) = 5, plot (2, 5). Connect (1, 3) through (2, 5) to (4, 9) with a straight segment; do not extend beyond x = 1 or x = 4.
Graph f(x) = x + 2 for the domain [0, 3]. State the endpoint coordinates and dot styles.
Closed dot at (0, 2); closed dot at (3, 5); straight segment connecting them.
Why: Both brackets mean both endpoints are included. f(0) = 2 and f(3) = 5. The function is linear, so the segment is straight.
Check: Recompute f(0) and f(3); confirm both dots are filled and the segment does not extend past x = 0 or x = 3.
Graph f(x) = x^2 for the domain (1, 3]. State endpoint coordinates and dot styles, and give one interior point.
Open dot at (1, 1); closed dot at (3, 9); interior point (2, 4); smooth curve connecting them.
Why: Parenthesis at 1 means open; bracket at 3 means closed. f(1) = 1, f(3) = 9, f(2) = 4. The curve is part of a parabola.
Check: Verify f(1) = 1 and f(3) = 9; confirm open dot at x = 1 and closed dot at x = 3; check that (2, 4) lies on the plotted curve.
Graph f(x) = 3 - x for the domain [2, 5). State endpoint coordinates and dot styles.
Closed dot at (2, 1); open dot at (5, -2); straight segment connecting them.
Why: Bracket at 2 means closed; parenthesis at 5 means open. f(2) = 1 and f(5) = -2. Linear function gives a straight segment.
Check: Recompute f(2) = 1 and f(5) = -2; confirm closed dot at x = 2 and open dot at x = 5; ensure no extension beyond endpoints.
Graph f(x) = 2x - 1 for the domain (-2, 2). State endpoint coordinates and dot styles, and give one interior point.
Open dot at (-2, -5); open dot at (2, 3); interior point (0, -1); straight segment connecting them.
Why: Both parentheses mean both endpoints are excluded. f(-2) = -5, f(2) = 3, f(0) = -1. Linear function gives a straight segment.
Check: Verify f(-2) = -5 and f(2) = 3; confirm both dots are unfilled; check that (0, -1) lies on the segment and the graph stops at x = -2 and x = 2.
The endpoint x-value is included in the domain, so the function value at that x is part of the graph.
A parenthesis next to an endpoint means open; a bracket means closed.
No. Plot endpoints and enough interior points to show the shape accurately.
Yes. Mixed notation such as [1, 4) is common and requires one closed and one open dot.