Plot it point by point and a surprise appears: two separate branches that hug the axes but never touch — a hyperbola.
We know the rule y = k∕x; now let's see it. The honest way to meet a new graph is the oldest one: make a table, plot the points, and connect them. The shape that appears is unlike any line you've drawn — two separate curving pieces that swoop in toward the axes and flatten out along them, never quite arriving. That shape has a name, the hyperbola, the two axes are its asymptotes, and the sign of k decides which corners of the plane the two branches call home. Everything in this lesson grows from one table of values for y = 6∕x.
The surest way to learn a graph's shape is to compute a handful of points and let the picture emerge. Take y = 6∕x and choose x-values that divide 6 evenly — the arithmetic stays clean. For x = 1, 2, 3, 6 we get y = 6, 3, 2, 1, and the negatives mirror them.
| x | 1 | 2 | 3 | 6 | −1 | −2 | −3 | −6 |
|---|---|---|---|---|---|---|---|---|
| y = 6∕x | 6 | 3 | 2 | 1 | −6 | −3 | −2 | −1 |
Plot the pairs (1, 6), (2, 3), (3, 2), (6, 1), … and a smooth curve all but draws itself. Each point checks out by the product: 1·6 = 6, 2·3 = 6, 3·2 = 6 — every plotted point has x·y = 6.
To graph y = k∕x: pick convenient x-values (factors of k are easiest), compute y = k∕x for each, plot the pairs, and connect them with a smooth curve — never a straight ruler.
Drag the slider to pick an x; the amber point lands exactly on the curve at (x, 6∕x). Near x = 0 the output shoots off — that gap is why the picture comes in two pieces.
Here is the first surprise. Connecting the points does not give one line, the way y = kx + b always did. It gives two separate curving branches — one in the upper right where x and y are both positive, and a matching one in the lower left where both are negative. This double curve has a name: it is a hyperbola.
The two branches are genuinely not joined. There is no point at x = 0 to bridge them, because y = 6∕0 has no value — that is the forbidden slit at the origin we met when we defined inverse proportion. So a pencil tracing the curve from the right branch can never cross over to the left without lifting off the page.
A hyperbola comes in two branches, one on each side of the missing x = 0. The branches are separate curves — they are not connected.
Look at what each branch does as you travel along it. Push x toward 0 — say x = 1, then 0.5, then 0.1 — and y = 6∕x climbs to 6, 12, 60, shooting up without bound. Push x the other way — x = 6, 60, 600 — and y sinks to 1, 0.1, 0.01, sliding toward 0 but never reaching it.
So each branch hugs the y-axis near the slit and hugs the x-axis far out, getting infinitely close but never touching. A line that a curve approaches like this is called an asymptote, and for y = k∕x the asymptotes are exactly the two axes — already drawn on every grid.
"Never touches" is exact, not approximate. y = 6∕x equals 0 for no value of x (a fraction is zero only when its top is zero, and the top is 6), and it is undefined at x = 0. So the curve meets neither axis.
Which corners of the plane do the branches live in? The product form settles it. Since x·y = k, if k > 0 then x and y must have the same sign — their product is positive only when both are positive or both are negative.
Both positive lands you in Quadrant Ⅰ (top right); both negative lands you in Quadrant Ⅲ (bottom left). So a k > 0 hyperbola sits in Ⅰ and Ⅲ. For y = 6∕x: as x rises through positive values y falls but stays positive (the Ⅰ branch); the Ⅲ branch is its mirror through the origin.
k > 0 ⇒ the branches sit in Quadrants Ⅰ and Ⅲ, where x and y share a sign.
Step k through positive and negative values. The branches jump between Ⅰ & Ⅲ and Ⅱ & Ⅳ — but the shape never changes.
Flip the sign of k and the same reasoning runs backward. If k < 0, then x·y = k is negative, so x and y must have opposite signs. That puts the branches in Quadrant Ⅱ (x < 0, y > 0, top left) and Quadrant Ⅳ (x > 0, y < 0, bottom right).
Worked example: y = −6∕x. At x = 2, y = −3 (Quadrant Ⅳ); at x = −2, y = 3 (Quadrant Ⅱ). Step the slider above into the negatives to see exactly this. The shape of the curve is identical to the k > 0 case — only which two quadrants it occupies has changed.
| sign of k | same / opposite sign | quadrants |
|---|---|---|
| k > 0 | x and y have the same sign | Ⅰ & Ⅲ |
| k < 0 | x and y have opposite signs | Ⅱ & Ⅳ |
The hyperbola's shape is the same for every k ≠ 0. Only the sign of k changes which pair of quadrants the branches occupy — never confuse "shape" with "location."
On which inverse proportion does the point (−2, 3) lie? Test the product: (−2)(3) = −6, so k = −6 and the point is on y = −6∕x — a k < 0 curve through Quadrants Ⅱ and Ⅳ. It is not on y = 6∕x, whose points all have x·y = +6.
To graph y = k∕x you make a table, plot the pairs, and connect them — and out comes a hyperbola, not a line. It has two separate branches, one on each side of the forbidden slit at x = 0, and the branches never join. Each branch hugs both axes — getting infinitely close but never touching — so the x-axis and y-axis are the asymptotes. The sign of k places the branches: k > 0 puts them in Quadrants Ⅰ & Ⅲ (x and y share a sign), while k < 0 puts them in Ⅱ & Ⅳ (opposite signs). The shape is always the same; only the location moves. Next we read the properties of this curve straight off the picture.
Complete the table for y = 8∕x at x = 1, 2, 4, 8.
y = 8, 4, 2, 1. (Each is 8 ÷ x, and each pair has x·y = 8.)
Which quadrants does the graph of y = 5∕x occupy?
Quadrants Ⅰ and Ⅲ. Since k = 5 > 0, x and y must share a sign.
Which quadrants does the graph of y = −10∕x occupy?
Quadrants Ⅱ and Ⅳ. Since k = −10 < 0, x and y have opposite signs.
Does the graph of y = 3∕x ever touch the x-axis? Explain.
No. It approaches the x-axis but never reaches it — 3∕x equals 0 for no value of x. The axes are the curve's asymptotes.
A hyperbola has how many branches?
Two. They sit on opposite sides of the missing point x = 0 and never join.
The point (−2, 3) lies on which inverse proportion: y = 6∕x or y = −6∕x?
y = −6∕x, since (−2)(3) = −6, so k = −6. (On y = 6∕x every point has x·y = +6.)
Six questions to lock it in. Tap the answer you think is right.
This lesson turns the rule y = k∕x into a picture, and the picture carries the whole idea. The big idea is that an inverse proportion graphs as a hyperbola: two separate curving branches that approach the axes as asymptotes but never reach them, with the sign of k deciding the quadrant pair (k > 0 → Ⅰ & Ⅲ, k < 0 → Ⅱ & Ⅳ). This is the first graph in the course that is a curve rather than a line, and the first whose rate of change is not constant.
The misconceptions to watch are three. Students often connect the two branches across x = 0 with a stray segment — stress that there is no point there, so the branches cannot join. Many believe the curve eventually touches an axis — it gets arbitrarily close but never lands, because k∕x is never 0 and is undefined at x = 0. And the sign rule gets flipped — anchor it to the product: x·y = k, so a positive k forces matching signs (Ⅰ & Ⅲ) and a negative k forces opposite signs (Ⅱ & Ⅳ).
This material aligns with US Common Core standards 8.F.A.3 and F-IF.C.7 (graph a function and interpret its key features — here a curve, not a line), F-IF.B.4 and F-IF.B.5 (domain and end behavior, including asymptotic behavior), and 8.F.B.5 (describe a graph qualitatively).