Graphs of reciprocal and rational functions

Graph of y = 1/x

The reciprocal graph, a rectangular hyperbola: two branches, in the first and third quadrants, that approach both axes without touching them.

Every point on the curve has xy = 1, so y = 1/x is the graph of inverse proportion: doubling x halves y. Points to plot: (0.5, 2), (1, 1), (2, 0.5) on the right-hand branch and (−0.5, −2), (−1, −1), (−2, −0.5) on the left.

The gradient is dy/dx = −1/x², which is negative for every x ≠ 0, so the function is decreasing on each branch (x < 0 and x > 0). The points of each branch closest to the origin are (1, 1) and (−1, −1), at a distance of √2. Many curves you sketch in exams are this one transformed, such as y = 3/(x − 1) + 2.

Key features

It is its own inverse: reflecting it in y = x gives the same graph.

It has two lines of symmetry, y = x and y = −x.

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 0 (the y-axis); y = 0 (the x-axis).
2. Only draw the curve where it is defined: x ≠ 0.
3. Check the range: y ≠ 0. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

Graphs made from this one: y = −1/x, y = 1/(x − 2) + 3, y = (2x + 1)/(x − 1). Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = −1/x

The reciprocal graph reflected in the x-axis: its branches are in the second and fourth quadrants.

Every point on the curve has xy = −1, so when x is positive y is negative and the other way round. Points to plot: (0.5, −2), (1, −1), (2, −0.5) on the right-hand branch and (−0.5, 2), (−1, 1), (−2, 0.5) on the left.

The gradient is dy/dx = 1/x², which is positive for every x ≠ 0, so the function is increasing on each branch. Because −1/x = 1/(−x), the same graph is also y = 1/x reflected in the y-axis. Like y = 1/x it is its own inverse, with lines of symmetry y = x and y = −x.

Key features

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 0 (the y-axis); y = 0 (the x-axis).
2. Only draw the curve where it is defined: x ≠ 0.
3. Check the range: y ≠ 0. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

y = −1/x is y = 1/x reflected in the x-axis: see the graph of y = 1/x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = 1/x²

Both branches are above the x-axis, because x² is never negative. The graph is symmetric about the y-axis.

Key features

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 0 (the y-axis); y = 0 (the x-axis).
2. Only draw the curve where it is defined: x ≠ 0.
3. Check the range: y > 0. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = 1/(x − 2) + 3

The graph of y = 1/x moved 2 right and 3 up, so the asymptotes move to x = 2 and y = 3.

Key features

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 2; y = 3.
2. Mark the x-intercept (5/3, 0), the y-intercept (0, 5/2).
3. Only draw the curve where it is defined: x ≠ 2.
4. Check the range: y ≠ 3. Your sketch should not go anywhere else.

y = 1/(x − 2) + 3 is y = 1/x translated 2 right and 3 up: see the graph of y = 1/x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = (2x + 1)/(x − 1)

A rational function of the form (ax + b)/(cx + d). Dividing gives 2 + 3/(x − 1): the graph of y = 1/x stretched by 3, moved 1 right and 2 up.

Key features

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 1; y = 2.
2. Mark the x-intercept (−1/2, 0), the y-intercept (0, −1).
3. Only draw the curve where it is defined: x ≠ 1.
4. Check the range: y ≠ 2. Your sketch should not go anywhere else.

y = (2x + 1)/(x − 1) is y = 1/x stretched vertically by scale factor 3, then translated 1 right and 2 up: 3/(x − 1) + 2: see the graph of y = 1/x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x + 1/x

A graph with a vertical asymptote at x = 0 and an oblique (slant) asymptote y = x. Setting dy/dx = 1 − 1/x² = 0 gives turning points at x = ±1.

Key features

How to sketch it

1. Draw the asymptotes first as dashed lines: x = 0 (the y-axis); y = x.
2. Mark the turning points (−1, −2) and (1, 2).
3. Only draw the curve where it is defined: x ≠ 0.
4. Check the range: y ≤ −2 or y ≥ 2. Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = 1/(x² + 1)

A bell-like curve with no vertical asymptote (x² + 1 is never 0). Its highest point is (0, 1) and it approaches the x-axis on both sides.

Key features

How to sketch it

1. Draw the asymptote first as dashed lines: y = 0 (the x-axis).
2. Mark the y-intercept (0, 1), the turning point (0, 1).
3. The domain is every real number, so the curve continues off both sides of your axes.
4. Check the range: 0 < y ≤ 1. Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.