HomeDrawing Reciprocal Graphs with Asymptotes: A Step-by-Step Worksheet Guide Free original lessons
Drawing Reciprocal Graphs with Asymptotes: A Step-by-Step Worksheet Guide
Learn to plot y = a/(x-h)+k from value tables, mark vertical and horizontal asymptotes, and check branch behavior with clear examples and practice.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Reciprocal graphs of the form y = a/(x - h) + k have two separate branches and two asymptotes: a vertical line x = h and a horizontal line y = k. This guide shows how to build a value table, plot points on both sides of the vertical asymptote, and draw the branches so they approach the asymptotes without crossing them. The method is arithmetic-based: choose x-values near and away from h, compute y, and use symmetry and sign patterns to confirm the shape. All numerical values used in examples are hypothetical and chosen for easy checking; measure your own grid and equipment if you produce printed materials.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Identify the asymptotes before plotting
For y = a/(x - h) + k, the denominator is zero at x = h, so the vertical asymptote is x = h. As x grows large positive or negative, a/(x - h) approaches 0, so the horizontal asymptote is y = k. Mark both as dashed lines on the grid first; they are guides, not part of the curve.
The sign of a controls which quadrants the branches occupy relative to the asymptotes. If a > 0, the branch on the right of x = h is above y = k and the branch on the left is below y = k. If a < 0, the pattern reverses. This sign check prevents plotting errors before any points are computed.
- Write the equation in the form y = a/(x - h) + k and read off h and k.
- Draw dashed lines at x = h and y = k on the coordinate grid.
- Note the sign of a to predict whether each branch lies above or below the horizontal asymptote.
- Label the asymptotes clearly so the reader can distinguish them from the curve.
Build a value table near and away from the vertical asymptote
Choose x-values on both sides of h. Include values close to h, such as h - 1, h - 0.5, h + 0.5, and h + 1, and values farther away, such as h - 3 and h + 3. Compute y for each using simple arithmetic. Keep units consistent and record exact fractions or decimals to two places.
The table should show the characteristic behavior: as x approaches h from either side, the magnitude of y grows large; as x moves away from h, y approaches k. Plotting these points reveals the steep near-asymptote section and the flattening far section.
- List x-values in increasing order, skipping x = h.
- Substitute each x into y = a/(x - h) + k and compute y.
- Record whether y is above or below k and compare with the sign prediction.
- Plot each point lightly and connect points on the same side of x = h with a smooth curve.
Draw and check the two branches
Each side of the vertical asymptote forms one branch. Draw a smooth curve through the plotted points on that side, approaching the dashed asymptotes but never touching or crossing them. The branch should become steeper as it nears x = h and flatter as it moves away.
Check the drawing by selecting one x-value from the table and confirming the plotted point lies on the drawn curve. Also confirm that no branch crosses x = h or y = k. If a point appears on the wrong side of an asymptote, recheck the arithmetic and the sign of a.
- Connect points on the left of x = h with one smooth branch.
- Connect points on the right of x = h with a second smooth branch.
- Extend each branch toward the asymptotes using the trend in the table.
- Verify at least two plotted points and confirm neither branch crosses an asymptote.
Worked example
Draw the graph of y = 2/(x - 1) + 3. Mark the asymptotes and plot points for x = -1, 0, 0.5, 1.5, 2, and 4.
Worked answer
Vertical asymptote: x = 1. Horizontal asymptote: y = 3. Since a = 2 > 0, the right branch is above y = 3 and the left branch is below y = 3. Computed points: x = -1 gives y = 2/(-2) + 3 = 2; x = 0 gives y = 2/(-1) + 3 = 1; x = 0.5 gives y = 2/(-0.5) + 3 = -1; x = 1.5 gives y = 2/(0.5) + 3 = 7; x = 2 gives y = 2/(1) + 3 = 5; x = 4 gives y = 2/(3) + 3 = 3.67 (to two decimal places). Plot these points, draw dashed lines at x = 1 and y = 3, and connect the left points (-1, 2), (0, 1), (0.5, -1) with a smooth branch approaching x = 1 vertically and approaching y = 3 as x → -∞, and the right points (1.5, 7), (2, 5), (4, 3.67) with a second branch approaching x = 1 vertically and approaching y = 3 as x → +∞. Check: re-substitute x = 0.5 to confirm y = -1, and confirm the left-branch y-values are all below 3 while the right-branch y-values are all above 3; no point is plotted at x = 1.
Four practice exercises
Exercise 1
Draw y = 1/(x - 2) + 1. Mark the asymptotes and plot points for x = 0, 1, 1.5, 2.5, 3, and 5.
Show answer and check
Vertical asymptote x = 2; horizontal asymptote y = 1. Points: (0, 0.5), (1, 0), (1.5, -1), (2.5, 3), (3, 2), (5, 1.33). Left branch below y = 1, right branch above y = 1. Draw two separate smooth branches, one on each side of x = 2, approaching the dashed asymptotes without crossing them.
Why: Substitute each x into y = 1/(x - 2) + 1. Since a = 1 > 0, the right branch is above y = 1 and the left branch is below. The two branches approach x = 2 and y = 1 without crossing.
Check: Confirm each computed y by re-substituting, verify that left-branch y-values are less than 1 and right-branch y-values are greater than 1, and check that no plotted point lies at x = 2.
Exercise 2
Draw y = -2/(x + 1) + 4. Mark the asymptotes and plot points for x = -3, -2, -1.5, -0.5, 0, and 2.
Show answer and check
Vertical asymptote x = -1; horizontal asymptote y = 4. Points: (-3, 5), (-2, 6), (-1.5, 8), (-0.5, 0), (0, 2), (2, 3.33). Left branch above y = 4, right branch below y = 4. Draw two separate smooth branches, one on each side of x = -1, approaching the dashed asymptotes without crossing them.
Why: Here h = -1 and k = 4. Since a = -2 < 0, the branch on the right of x = -1 is below y = 4 and the branch on the left is above y = 4. Compute each y carefully, noting that x + 1 is negative for x < -1.
Check: Recompute one point on each side, confirm left-branch y-values exceed 4 and right-branch y-values are less than 4, and verify the branches do not cross x = -1 or y = 4.
Exercise 3
Draw y = 3/(x - 0) - 2, that is y = 3/x - 2. Mark the asymptotes and plot points for x = -3, -1, -0.5, 0.5, 1, and 3.
Show answer and check
Vertical asymptote x = 0 (the y-axis); horizontal asymptote y = -2. Points: (-3, -3), (-1, -5), (-0.5, -8), (0.5, 4), (1, 1), (3, -1). Left branch below y = -2, right branch above y = -2. Draw two separate smooth branches, one on each side of the y-axis, approaching the dashed asymptotes without crossing them.
Why: With h = 0 and k = -2, the vertical asymptote is the y-axis. Since a = 3 > 0, the right branch is above y = -2 and the left branch is below. The values near x = 0 have large magnitude, showing the steep approach.
Check: Check that x = 0 is not plotted, confirm the sign pattern relative to y = -2, and re-substitute two points to verify arithmetic.
Exercise 4
Draw y = -1/(x - 3) + 0, that is y = -1/(x - 3). Mark the asymptotes and plot points for x = 1, 2, 2.5, 3.5, 4, and 6.
Show answer and check
Vertical asymptote x = 3; horizontal asymptote y = 0 (the x-axis). Points: (1, 0.5), (2, 1), (2.5, 2), (3.5, -2), (4, -1), (6, -0.33). Left branch above y = 0, right branch below y = 0. Draw two separate smooth branches, one on each side of x = 3, approaching the dashed asymptotes without crossing them.
Why: Here k = 0, so the horizontal asymptote is the x-axis. Since a = -1 < 0, the right branch is below y = 0 and the left branch is above y = 0. Compute each y and note the sign change across x = 3.
Check: Verify that left-branch y-values are positive and right-branch y-values are negative, confirm no point lies at x = 3, and re-substitute one point from each branch.
Common mistakes
- Plotting a point at x = h by substituting into the equation. The expression is undefined at x = h. Mark x = h only as a dashed vertical asymptote and never place a curve point there.
- Drawing one continuous curve through both sides of the vertical asymptote. Reciprocal graphs have two separate branches. Draw one branch on each side of x = h and leave a gap at the asymptote.
- Letting a branch cross the horizontal asymptote y = k. The curve approaches y = k but does not cross it. Check that all plotted y-values on one side are consistently above or below k.
- Ignoring the sign of a when predicting branch positions. Use the sign of a to decide whether the right branch is above or below y = k, then confirm with computed points.
Self-check rubric
- Asymptotes correctly identified and drawn as dashed lines at x = h and y = k.
- Value table includes at least two x-values on each side of h, with correct arithmetic.
- Two separate branches drawn, one on each side of the vertical asymptote, without crossing either asymptote.
- Branch positions match the sign of a relative to y = k.
- At least two plotted points visibly lie on the drawn curves.
- Labels distinguish asymptotes from the curve; alternative valid point choices and smooth curve styles are accepted if the asymptote behavior and sign pattern are correct.
Questions and answers
Why does the graph have two separate branches?
The denominator x - h is zero at x = h, so the function is undefined there. The curve splits into a left branch and a right branch, each approaching the vertical asymptote.
How close should the x-values be to the vertical asymptote?
Include at least one value within 1 unit of h on each side, such as h - 0.5 and h + 0.5, and one value farther away, such as h - 3. Closer values make the steep approach visible.
Can the horizontal asymptote be at y = 0?
Yes. If k = 0, the horizontal asymptote is the x-axis. The same plotting method applies; just mark y = 0 as the dashed horizontal guide.
How do I check my graph without an answer key?
Pick two x-values from your table, confirm their plotted points lie on your drawn branches, and verify that no branch crosses either asymptote. Also check that the sign of y - k matches the sign of a/(x - h).