What an asymptote is
A bottle of juice from the fridge keeps warming, but it never gets warmer than the room. What is its graph heading toward?
Where you are 10 in Features of a graph
Why this exists
A bottle of juice comes out of the fridge at 6^\circ\text{C} into a room at 22^\circ\text{C}. It warms quickly at first, then more and more slowly. It never gets warmer than the room, yet if you wait long enough it gets as close to room temperature as you like. Its graph does not stop at a last point, and it does not run off the top or bottom of the page. It flattens out along the line at height 22.
A phone charging toward full behaves the same way, and so does the distance left in a walk that always covers half of what remains. Each graph is heading toward a line it gets ever closer to. Such a line is called an asymptote, from a Greek word meaning “not falling together”. Naming it is the quickest way to say where a graph is heading far out, and with end behavior it is one of the features of a graph worth reading off.
Definition
A horizontal asymptote of a graph is a horizontal line y = k that the graph gets as close to as you like as x grows without bound, or as x is negative and grows in size without bound.
“As close as you like” carries the meaning. Name any distance, however small: 0.1, 0.001, a millionth. Far enough out, every point of the graph is within that distance of the line, and it stays within it from then on. A graph that passes near k once and moves away again, or that hovers a fixed distance from k, does not have y = k as an asymptote.
In the arrow notation of End behavior, y = k is a horizontal asymptote on the right when
and on the left when the same holds as x \to -\infty. A graph may have one on the right, one on the left, one on each side, or none.
The definition does not say the graph never reaches the line. Every graph in this lesson stays strictly on one side of its asymptote and never touches it. But later courses meet graphs that cross their horizontal asymptote, even many times, on their way to settling down. The line says where the graph is heading, not a wall it cannot pass. Nor does the definition say anything about the middle of the graph: only the far ends count.
An asymptote in general is any line a graph approaches in this way. This lesson is about the horizontal kind; the vertical kind appears briefly under Boundaries.
Intuition
Dominika stands 8 meters from a wall. With each step she walks half the distance still left.
| Steps taken, n | Distance left, d (m) |
|---|---|
| 0 | 8 |
| 1 | 4 |
| 2 | 2 |
| 3 | 1 |
| 4 | \displaystyle \frac{1}{2} |
| 5 | \displaystyle \frac{1}{4} |
| 6 | \displaystyle \frac{1}{8} |
Two facts pull against each other. The distance is never 0: half of a positive number is still positive, so after any number of steps some distance is left. And the distance gets as small as you like: after 10 steps it is \displaystyle \frac{8}{1024} m, under a centimeter, and after 20 steps it is under a hundredth of a millimeter. Plotted, the points drop quickly and then crowd along the horizontal axis. The line d = 0 is the asymptote.
Steps come in whole numbers, so the picture is dots rather than a curve, but the idea is the same. It also shows what really shrinks: the gap between the graph and the line. For the walk the gap is the distance to the wall. For the juice it is 22 - T, the degrees below room temperature. An asymptote is a statement that a gap shrinks toward 0.
Results
A base between 0 and 1, read to the right. If a > 0 and 0 < b < 1, the graph of y = a \cdot b^x has the horizontal asymptote y = 0 on the right, and lies above it.
A base greater than 1, read to the left. If a > 0 and b > 1, the graph of y = a \cdot b^x has the horizontal asymptote y = 0 on the left, and lies above it.
Adding a constant. Adding k to every output moves the target from 0 to k: the graph of y = a \cdot b^x + k has the asymptote y = k on the same side, and lies above it.
At most one on each side. A graph cannot have two different horizontal asymptotes on the same side.
Why it is true
Each step of 1 to the right multiplies a \cdot b^x by b; these are the growth and decay factors. Take \displaystyle b = \frac{1}{2} first. Since 2^{10} = 1024, every 10 steps divide the output by more than 1000, so every 10 more steps shrink it below a thousandth of what it was. Whatever distance you name, enough rounds of 10 steps take the output below it. The output also stays positive, being a product of positive numbers. That is exactly the asymptote y = 0, with the graph above it.
For another base between 0 and 1, some fixed number of steps at least halves the output. For b = 0.85, (0.85)^5 \approx 0.444, so every 5 steps at least halve it; for b = 0.99 it takes 69 steps, since (0.99)^{69} \approx 0.4998. The halving argument then runs as before, only more slowly. Read to the left, a base b > 1 divides by b at each step, which is multiplying by \displaystyle \frac{1}{b}, a number between 0 and 1, so the same argument applies.
For y = a \cdot b^x + k, the gap between the graph and the line y = k is
which shrinks toward 0 and stays positive, by the first two results.
Finally, suppose the outputs settled toward both k_1 and k_2 on the same side, with k_1 < k_2. Name the distance \displaystyle \frac{1}{3}(k_2 - k_1). Far enough out, every output would be within that distance of both numbers at once, and no number is: the two windows around k_1 and k_2 do not overlap.
Worked examples
Reading an asymptote from a table
Make a table for y = 3 \cdot (0.5)^x and use it to find any horizontal asymptote.
SolutionTake inputs that move out to the right, plus one far to the left.
| x | y |
|---|---|
| -10 | 3072 |
| 0 | 3 |
| 1 | 1.5 |
| 2 | 0.75 |
| 5 | 0.09375 |
| 10 | 0.00293 |
| 20 | 0.0000029 |
Going right, each step of 1 halves the output, so the outputs shrink toward 0. They never reach it: each is 3 times a positive number. So y = 0 is a horizontal asymptote on the right, with the graph above it.
Going left, each step doubles the output: y = 3 \cdot 2^{10} = 3072 at x = -10. The outputs grow without bound, so there is no asymptote on the left.
An asymptote on the left
Find any horizontal asymptote of y = 2^x + 3 from a table.
SolutionEach output is 3 more than 2^x. Going left, 2^x halves at every step (1, \displaystyle \frac{1}{2}, \displaystyle \frac{1}{4}, ...), shrinking toward 0 but staying positive.
| x | y |
|---|---|
| 10 | 1027 |
| 0 | 4 |
| -1 | 3.5 |
| -2 | 3.25 |
| -5 | 3.03125 |
| -10 | 3.00098 |
| -20 | 3.00000095 |
The outputs crowd toward 3 from above, and the gap y - 3 = 2^x is never 0. So y = 3 is a horizontal asymptote on the left. Going right, 2^{10} + 3 = 1027, and 2^x doubles at every step, so the outputs grow without bound: no asymptote on the right.
Reading from a graph: a phone charging
Henrik plugs in his phone at 10 percent. The graph shows its charge P, in percent, t hours later. What line is the graph heading toward, and what does it mean? The graph was drawn from the made-up model P = 100 - 90(0.3)^t; use a table to confirm the reading.
The curve starts at the closed dot (0, 10), climbs steeply, then bends and runs just under the height 100, flattening as it goes. Only the right end exists, since t \geq 0. The reading: the graph is heading toward the line P = 100 on the right, from below.
| t (hours) | P (%) |
|---|---|
| 0 | 10 |
| 1 | 73 |
| 2 | 91.9 |
| 3 | 97.57 |
| 4 | 99.271 |
| 6 | 99.934 |
The formula says why. The charge still missing is 100 - P = 90(0.3)^t, and each hour multiplies it by 0.3. So each hour fills 70\% of whatever was missing: a lot when much is missing, very little near full. The missing part shrinks toward 0 and stays positive, so P = 100 is the asymptote: the phone approaches full charge.
The juice warming up
The juice’s temperature, t minutes after it leaves the fridge, is given by the made-up model T = 22 - 16(0.9)^t degrees Celsius. (a) Find the starting temperature and the horizontal asymptote, and say what the asymptote means. (b) Find the first whole minute at which the juice is within 1^\circ\text{C} of the asymptote, and the first at which it is within 0.1^\circ\text{C}.
(a) At t = 0, (0.9)^0 = 1, so T = 22 - 16 = 6^\circ\text{C}. The gap below 22 is 22 - T = 16(0.9)^t. Each minute multiplies it by 0.9, so it shrinks toward 0 while staying positive. The asymptote is T = 22: the juice approaches room temperature, always a little below it.
(b) Within 1^\circ\text{C} means the gap 16(0.9)^t is under 1. Trying whole minutes:
| t | 26 | 27 | 48 | 49 |
|---|---|---|---|---|
| gap 22 - T | 1.034 | 0.930 | 0.102 | 0.092 |
At 26 minutes the juice is still 1.034^\circ below the room, and at 27 minutes it is within 1^\circ. Within 0.1^\circ first happens at 49 minutes. Each closer target is met, only later: that is “as close as you like” in action.
How close the walk gets
Dominika’s distance from the wall after n steps is d = 8(0.5)^n meters. After how many steps is she within 1 centimeter of the wall? Within 1 millimeter?
SolutionThe distance is \displaystyle \frac{8}{2^n} meters. Under 1 cm, which is 0.01 m, needs 2^n > 800. Since 2^9 = 512 and 2^{10} = 1024, the first is n = 10, leaving \displaystyle \frac{8}{1024} \approx 0.0078 m, about 0.78 cm.
Under 1 mm, 0.001 m, needs 2^n > 8000. Since 2^{12} = 4096 and 2^{13} = 8192, it takes n = 13 steps.
Each further factor of 10 in closeness costs 3 or 4 more steps, because 2^3 = 8 < 10 < 16 = 2^4. Any closeness can be reached with enough steps, yet no number of steps leaves a distance of 0.
Boundaries
The inputs must go on forever. An asymptote describes a far end. If the inputs stop, as time does before the juice left the fridge, there is nothing to describe on that side. Reading where a graph starts and stops is Domain and range from a graph.
Crossing is allowed in general. The graphs y = a \cdot b^x + k of this lesson never touch their asymptote, because the gap a \cdot b^x is never 0. Other graphs can cross the line, for example one that swings above and below it in smaller and smaller waves as it settles. The line is still an asymptote.
A model is not the situation. The phone formula never reaches 100, but a phone that rounds to whole percentages displays 100 once the true value passes 99.5, which this model first does at about 4.3 hours. The asymptote describes the formula; how much of it is real depends on the situation.
A table or picture is evidence, not proof. A graph that rises very slowly can look level in a window (see End behavior). Certainty comes from a reason, such as “the gap is multiplied by 0.9 each minute”.
Vertical asymptotes. An asymptote can also be vertical: the graph of \displaystyle y = \frac{1}{x} shoots upward alongside the vertical axis as x shrinks toward 0 from the positive side, so the line x = 0 is a vertical asymptote, studied in Vertical asymptotes.
Where this goes next
The asymptote is the first landmark drawn when sketching y = a \cdot b^x + k in Graphs of exponential functions. Horizontal asymptotes finds them for other families of functions, Vertical asymptotes treats the vertical kind, and Asymptotes adds slanted ones. Much later, the phrase “as close as you like” is made exact (Limits at infinity and horizontal asymptotes).
Exercises
From tables
- A function has outputs 6.9, 6.99 and 6.999 at x = 10, 100 and 1000. What horizontal asymptote does the table suggest, on which side, and is the graph above or below it? How far from the line is the output at x = 1000?
- Complete a table for y = 5(0.2)^x at x = -2, 1, 2, 3 and 5, and find any horizontal asymptote.
- Work out y = 3^x - 4 at x = -1, -2, -3 and -6, to three decimal places, and find the horizontal asymptote and its side.
- Work out y = 10 + 2(0.5)^x at x = 1, 5 and 10. Name the horizontal asymptote and explain why the graph never reaches it.
From graphs
For each graph, give the equation of any horizontal asymptote, its side, and whether the graph is above or below it. If there is none, say why.
- Graph A.
- Graph B.
- Graph C.
- Graph D.
In situations
- Isobel does the halfway walk starting 12 meters from a wall. Give the distance left after 1, 2, 3 and 4 steps, and find the first step after which she is within 10 cm of the wall.
- A jug of water kept in a cellar is carried out into a warmer garden. Its temperature after t minutes is T = 30 - 20(0.9)^t degrees Celsius. Give the starting temperature, the asymptote and what it means, and T after 10 minutes to two decimal places.
- Henrik’s phone charges as P = 100 - 90(0.3)^t, with t in hours. After how many whole hours does it first show at least 99 percent? By what factor does the missing charge shrink each hour?
- A table has outputs 2, 2.5 and 2.75 at x = 1, 2 and 3, and the gap to 3 halves at every step. Give the next two outputs and the asymptote, and check that y = 3 - 2(0.5)^x fits all three given rows.
- Write a function of the form y = a \cdot b^x + k whose graph passes through (0, 3) and has the horizontal asymptote y = -2 on the right, with the graph above it.
Explaining
- Henrik says that after 100 steps of the halfway walk the distance left is “basically zero”, so Dominika has reached the wall. Explain what is wrong, and why the walk still has the wall as its asymptote.
- Fridolin says y = 2^x + 3 has the asymptote y = 3 on both sides. Show with one value that he is wrong.
- Explain why a graph cannot have two different horizontal asymptotes on the same side.
- Isobel says “an asymptote is a line the graph can never touch”. Say what is right about this for the graphs of this lesson, what is wrong in general, and describe a graph that crosses its asymptote.
- Calixto’s thermometer reads to a tenth of a degree, and 60 minutes after the juice left the fridge it reads 22.0^\circ\text{C}. Ettore says that proves the juice reached room temperature, so the model is wrong. Use T = 22 - 16(0.9)^t to settle it.
Answers
- y = 7 on the right, with the graph below it. The output at x = 1000 is 0.001 below the line; the gaps 0.1, 0.01, 0.001 shrink by a factor of 10 each time.
- The outputs are 125, 1, 0.2, 0.04 and 0.0016. Each step right multiplies by 0.2, so y = 0 is an asymptote on the right, graph above. Going left the outputs grow without bound, so there is none on the left.
- -3.667, -3.889, -3.963, -3.999. Going left 3^x shrinks toward 0 and stays positive, so the asymptote is y = -4 on the left, with the graph just above it.
- 11, 10.0625 and about 10.002. The asymptote is y = 10 on the right. The gap 2(0.5)^x is a product of positive numbers, so it is never 0 and the graph stays above the line.
- y = -1 on the left, graph above it. There is none on the right, where A rises without bound.
- y = 0 on the right, graph above it. None on the left.
- y = 3 on the left, graph below it. None on the right, where C falls without bound.
- None. D is a line with nonzero slope, so its outputs change by the same amount at every step and rise or fall without bound at both ends.
- 6, 3, 1.5 and 0.75 meters. After 6 steps \displaystyle \frac{12}{64} = 0.1875 m is left; after 7 steps, \displaystyle \frac{12}{128} = 0.09375 m, under 10 cm. So the first is step 7.
- 10^\circ\text{C}. The asymptote is T = 30: the water approaches the garden’s temperature of 30^\circ\text{C} from below. T(10) = 30 - 20(0.9)^{10} \approx 23.03^\circ\text{C}.
- 4 hours: P(3) = 97.57 and P(4) \approx 99.27. The missing charge 90(0.3)^t is multiplied by 0.3 each hour.
- 2.875 and 2.9375; the asymptote is y = 3 on the right, graph below. The formula gives 3 - 1 = 2, 3 - 0.5 = 2.5 and 3 - 0.25 = 2.75.
- For example y = 5(0.5)^x - 2. The asymptote y = -2 fixes k = -2, a base between 0 and 1 puts it on the right, and a + k = 3 at x = 0 gives a = 5, which is positive, so the graph lies above the line. Any base between 0 and 1 works.
- The distance after 100 steps is \displaystyle \frac{8}{2^{100}} meters: tiny, but a positive number, since halving a positive number never gives 0. The wall is still the asymptote because the definition asks only that the distance get as close to 0 as you like and stay there, which it does; it never asks the walk to arrive.
- At x = 10, y = 2^{10} + 3 = 1027, and the outputs keep growing to the right. The asymptote y = 3 is on the left only.
- Suppose the outputs settled toward k_1 and k_2 on one side, with k_1 < k_2. Name the distance \displaystyle \frac{1}{3}(k_2 - k_1). Far enough out every output would have to be within that distance of both numbers, but the two windows do not overlap, so no output can be.
- For every graph y = a \cdot b^x + k here it is true: the gap a \cdot b^x is never 0. In general it is wrong, because the definition is about the gap shrinking far out, not about touching. A graph that waves above and below y = 2 in smaller and smaller swings crosses y = 2 again and again, yet gets as close to it as you like, so y = 2 is its asymptote.
- T(60) = 22 - 16(0.9)^{60} \approx 21.971, which rounds to 22.0 on a thermometer reading to a tenth. The model and the thermometer agree: the juice is about 0.029^\circ below the room, too little for the thermometer to show. The reading does not prove the juice reached 22.