End behavior
A graph runs off both edges of the page. Where does it go after that?
Where you are 10 in Features of a graph
When you reach for it
Every drawn graph is a window. The page stops at its edges, but the inputs do not: there is always a larger x to the right and a more negative x to the left. Does a quantity keep growing out there? Does it shrink away to nothing? Does it settle at some steady value?
The end behavior of a graph answers those questions. It describes where the graph goes at its two ends: far to the left, where x is negative and very large in size, and far to the right, where x is positive and very large.
Reach for it whenever you describe the overall shape of a function. With the intercepts, where the graph rises and falls and its turning points, the ends are one of the key features of a graph. It is the same kind of description as in Graphs that tell a story, aimed at the two ends. It also tells families apart quickly, because lines, parabolas and exponential curves each have their own kind of ends.
The procedure
Two ends, three things an end can do
Describe each end on its own. For every graph in this lesson, each end does one of three things.
- It rises without bound: going further out, the graph climbs and keeps climbing. Name any height, however large, and far enough out the graph passes it and stays above it.
- It falls without bound: the same, downward, past any depth you name.
- It levels off toward a value: going further out, the outputs get closer and closer to one number, and stay as close to it as you like. The graph flattens out along a horizontal line.
“Without bound” carries the meaning. A graph that climbs to 1000 and then flattens is not rising without bound; it is leveling off. And only the far ends count. Turning points, intercepts and bumps in the middle make no difference, because once you go further out than all of them, they are behind you.
Arrow notation
The symbol \infty, read “infinity”, is not a number; it stands for “beyond every number”. An arrow \to is read “goes to” or “approaches”.
| What happens | In words | In arrows |
|---|---|---|
| going right | x grows without bound | x \to \infty |
| going left | x is negative and grows in size without bound | x \to -\infty |
| the end rises | the outputs grow without bound | f(x) \to \infty |
| the end falls | the outputs fall without bound | f(x) \to -\infty |
| the end levels toward 3 | the outputs get closer and closer to 3 | f(x) \to 3 |
A full statement joins an input arrow to an output arrow. “As x \to \infty, f(x) \to \infty” is read aloud as “as x goes to infinity, f of x goes to infinity”, and it means: far to the right, the graph rises without bound. A complete description is two such statements, one for each end, for example
which says the graph falls without bound on the left and levels toward 2 on the right. When both ends do the same thing, one statement covers both: “as x \to \pm\infty, f(x) \to \infty”. For a graph given as y = \ldots, write y \to in place of f(x) \to. Notice that “x \to \infty” never says x equals anything. It describes a direction of travel, not a place you arrive at.
Reading the ends from a graph
Look at the far left and the far right of the picture, and nowhere else. An arrowhead on the end of a curve means the graph carries on in that direction past the edge of the window. For each end, ask: is it heading up, heading down, or flattening out along a horizontal line? If it is flattening, find the height of that line, often drawn dashed.
A picture is evidence, not proof: whoever drew it chose the window, and the arrowheads are a promise that the trend carries on. When the formula is known, you can test that promise.
Reading the ends from a table of large inputs
Put in inputs far out on each side, jumping by factors of ten: 10, 100, 1000 on the right and -10, -100, -1000 on the left. Inputs near 0 show the middle of the graph, which is exactly the part end behavior ignores. Then read each column of outputs:
- outputs with one sign, growing in size from row to row, mean that end rises (positive) or falls (negative) without bound;
- outputs crowding closer and closer to one number mean that end levels toward that number.
The ends of the graphs you know
Lines. Along the line y = mx + b, each step of 1 to the right changes y by the Slope m. If m > 0, then n steps to the right raise y by nm, which passes any height once n is large enough, and n steps to the left lower y by nm. So a line with positive slope falls on the left and rises on the right:
A negative slope reverses both ends. A line cannot level off unless it is level the whole way, because its steps never get smaller. That level case is the constant function y = b, where m = 0: as x \to \pm\infty, y \to b.
Parabolas. The graph of y = ax^2 + bx + c with a \neq 0 is a parabola. If it opens up (a > 0), both ends rise; if it opens down (a < 0), both ends fall. To see why, write the first two terms as a product,
Far out on either side, ax is so large in size that adding b cannot change its sign, so ax + b has the sign of ax. If a > 0 that is the sign of x itself, so x(ax + b) multiplies two numbers of the same sign: positive, and huge. If a < 0 the two factors have opposite signs, and the product is negative and huge. Adding the fixed number c changes nothing that far out.
Exponential curves. Take y = a \cdot b^x with a > 0, as in Exponential growth and decay. For growth, b > 1, each step of 1 to the right multiplies the output by b, so as x \to \infty, y \to \infty. Each step to the left divides by b instead, and the outputs shrink toward 0 but stay positive: as x \to -\infty, y \to 0. The graph flattens onto the x-axis without touching it. Decay, 0 < b < 1, is the mirror image: the left end rises without bound and the right end levels toward 0.
A horizontal line that one end of a graph levels toward is called a horizontal asymptote; Horizontal asymptotes studies them, and What an asymptote is says what an asymptote is in general.
Worked examples
A graph that turns twice
Describe the end behavior of the graph below, in words and in arrow notation.
Look only at the two edges. On the left, the arrowhead points up: going further left, the graph climbs. On the right, the arrowhead points down: going further right, the graph drops. The left end rises without bound and the right end falls without bound:
The three x-intercepts and the two turning points change nothing, because both turns sit between x = -2 and x = 2, and the ends are about what happens far beyond them.
Check. The graph was drawn from \displaystyle y = -\frac{1}{6}(x^3 - 10x). At x = 100 this gives \displaystyle -\frac{1}{6}(1{,}000{,}000 - 1000) = -166{,}500, and at x = -100 it gives 166{,}500: very large and negative far right, very large and positive far left, as the picture said.
A graph that levels off
The graph of \displaystyle g(x) = 2 - \left(\frac{1}{2}\right)^x is drawn below, with the dashed line y = 2. Describe its end behavior.
On the right, the curve flattens just under the dashed line. Values from the formula agree:
| x | g(x) |
|---|---|
| 2 | 1.75 |
| 5 | 1.97 |
| 10 | 1.999 |
The formula says why. The number \displaystyle \left(\frac{1}{2}\right)^x halves with each step to the right, so it shrinks toward 0, and g(x) is 2 minus that shrinking amount. Since \displaystyle \left(\frac{1}{2}\right)^x is always positive, g(x) stays just below 2.
On the left, the curve drops out of the bottom of the window. Going left, \displaystyle \left(\frac{1}{2}\right)^x doubles at every step, so g(x) is 2 minus a number growing without bound: g(-10) = 2 - 2^{10} = -1022. So
Only the right end levels off. The two ends of one graph need not behave alike.
Lines from their slopes
Describe the end behavior of each line without drawing it: (a) y = -4x + 7, (b) \displaystyle y = \frac{1}{2}x - 4, (c) y = 7.
Solution(a) The slope is -4, so every step of 1 to the right lowers y by 4, and every step to the left raises it by 4. Those changes add up without limit:
Large inputs agree: x = 1000 gives -4(1000) + 7 = -3993 and x = -1000 gives 4007.
(b) The slope \displaystyle \frac{1}{2} is positive. It is gentle, but 2000 steps to the right still raise y by 1000, and the steps never stop: as x \to -\infty, y \to -\infty, and as x \to \infty, y \to \infty. Check: x = 1000 gives 496 and x = -1000 gives -504. A small slope changes how fast the ends get there, never where they go.
(c) Every input gives 7, so the graph is the horizontal line at height 7: as x \to \pm\infty, y \to 7. The outputs equal 7 all along, and “levels toward 7” covers that case too.
A parabola from a table of large inputs
Make a table of q(x) = -2x^2 + 3x + 40 at x = \pm 10, \pm 100 and \pm 1000, and use it to describe the end behavior.
SolutionSubstitute each input. At x = 10, for example, q(10) = -2(100) + 30 + 40 = -130.
| x | q(x) |
|---|---|
| -1000 | -2{,}002{,}960 |
| -100 | -20{,}260 |
| -10 | -190 |
| 10 | -130 |
| 100 | -19{,}660 |
| 1000 | -1{,}996{,}960 |
Down each side the outputs are negative and grow larger in size, far faster than the inputs. Both ends fall without bound: as x \to \pm\infty, q(x) \to -\infty.
This fits the shape: the coefficient of x^2 is -2, so the parabola opens down. In the product x(-2x + 3), far right the first factor is positive and the second negative; far left, the reverse. Either way the product is negative and huge, and the 40 makes no difference. Near 0 the picture is different, since q(0) = 40 is positive: the middle says nothing about the ends.
Mold on a slice of bread
Vito measures a patch of mold on a slice of bread. Its area, t days after his first measurement, is modeled by A(t) = 2(1.5)^t square centimeters, where a negative t means a day before he first measured. The slice is 12 cm by 11 cm, an area of 132\ \text{cm}^2. Describe the end behavior of A, and say what each end means for the bread.
SolutionThe base 1.5 is greater than 1 and the starting value 2 is positive, so this is exponential growth: each day multiplies the area by 1.5.
| t | A(t) in \text{cm}^2 |
|---|---|
| -10 | 0.035 |
| -5 | 0.263 |
| 0 | 2 |
| 5 | 15.19 |
| 10 | 115.33 |
| 20 | 6650.51 |
Going forward the area grows without bound, and going back it shrinks toward 0 while staying positive:
That is the end behavior of the formula, and the bread disagrees with both ends. The patch cannot be larger than the slice: A(10) = 115.33 is already close to 132, and A(11) = 173.00 is larger than the whole slice, so the model holds only up to about day 10. And the left end says the patch was ever smaller but never absent, while the mold really started on some particular day. A formula’s end behavior describes the formula; the situation decides how much of it is real.
Boundaries
An end needs inputs that go on forever. If a graph stops, there is no end behavior on that side. In a situation the sensible inputs often stop: a value that is tracked from the day something was bought has no left end to describe. Reading where a graph starts and stops is Domain and range from a graph.
A picture cannot always tell slow rising from leveling off. The outputs of \displaystyle y = \sqrt{x} at x = 100, 1000 and 10{,}000 are 10, about 31.6 and 100. In a narrow window the graph looks as though it is leveling off, but it rises without bound. A table can mislead too, since it shows only a few rows. Certainty comes from a reason that holds for every large input, like the step argument for lines or the product argument for parabolas.
Rise, fall and level are not the only possibilities. Some graphs keep waving up and down forever without settling, such as the wave-shaped graphs of trigonometry. Their ends do none of the three things.
End behavior says nothing about the middle. A graph whose ends level toward a value may still touch or cross that value somewhere in the middle, and turns, bumps or even breaks in the middle leave the ends unchanged.
Where this goes next
An end that levels toward a horizontal line becomes Horizontal asymptotes, and the “one end rises, the other levels toward a line” picture is the backbone of Graphs of exponential functions. For polynomials such as x^3 - 4x, Degree, end behavior and the shape of a polynomial graph shows that the highest power alone decides both ends. Much later, the words “closer and closer” are made exact (Limits at infinity and horizontal asymptotes).
Exercises
Reading ends from a graph
For each graph, describe both ends in words and in arrow notation.
- Graph A.
- Graph B.
- Graph C.
- Graph D.
Words and arrows
- Write in words: as x \to -\infty, f(x) \to 5, and as x \to \infty, f(x) \to \infty.
- Write in arrow notation: “Far to the left the outputs fall without bound, and far to the right they get closer and closer to -1.”
- Saoirse-Ann’s function has as x \to \pm\infty, f(x) \to -\infty. Which of these could it be: a line with positive slope, a parabola opening up, a parabola opening down, or an exponential decay curve y = a \cdot b^x with a > 0 and 0 < b < 1? Explain.
- Sketch a graph that passes through (0, 1) with as x \to -\infty, f(x) \to 3 and as x \to \infty, f(x) \to -\infty. Then check that f(x) = 3 - 2 \cdot 2^x has all three features.
Lines, parabolas and exponential curves
Describe the end behavior of each graph in arrow notation.
- y = 5x - 11
- y = -0.2x + 300
- y = 3x^2 - 40x
- y = 10(0.9)^x
From tables of large inputs
- A function has f(-1000) = 4.002, f(-100) = 4.02, f(100) = 3.98 and f(1000) = 3.998. Describe both ends, including which side of the level each end is on.
- Wilda bought a commemorative coin for 60 dollars. Its collector value V(t), in dollars, t years later is V(0) = 60.00, V(5) = 27.73, V(10) = 17.15, V(20) = 12.55, V(40) = 12.01 and V(60) = 12.00. Describe the right end in words and in arrows, and say what it means for the coin. Why is there no left end to describe here?
- Ulla-Mai works out p(x) = x^2 - 10x + 32 at x = 3, 4 and 5 and gets 11, 8 and 7. Work out p at x = 10, 100, 1000, -100 and -1000, describe both ends, and say what her three rows were showing.
Explaining
- Explain why the graph of a line with a nonzero slope cannot level off at either end.
- Xerxes says that a graph whose ends both level toward 3 can never touch the line y = 3. Give an equation of a graph that proves him wrong, then describe a sketch of a different one that crosses the line y = 3.
Answers
- As x \to \pm\infty, y \to \infty. Graph A is a parabola opening up; both arrowheads point up.
- The left end rises without bound and the right end levels toward 0, flattening onto the x-axis. As x \to -\infty, y \to \infty; as x \to \infty, y \to 0.
- The left end levels toward -2, just above the dashed line, and the right end rises without bound. As x \to -\infty, y \to -2; as x \to \infty, y \to \infty.
- Both ends level toward 3, each from below. As x \to \pm\infty, y \to 3. The valley at the origin is in the middle and plays no part.
- Far to the left the outputs get closer and closer to 5; far to the right they grow without bound.
- As x \to -\infty, f(x) \to -\infty; as x \to \infty, f(x) \to -1.
- Only the parabola opening down, whose ends both fall. A line with positive slope falls on one end and rises on the other, a parabola opening up rises at both ends, and the decay curve rises on the left and levels toward 0 on the right.
- Draw the dashed line y = 3, let the far left of the curve run alongside it, pass through (0, 1), and drop off the bottom on the right. For the formula, f(0) = 3 - 2 \cdot 1 = 1. Going right, 2^x doubles each step, so 3 minus twice it falls without bound: f(10) = -2045. Going left, 2^x halves each step and stays positive, so f(x) sits just below 3: f(-10) \approx 2.998.
- Slope 5 > 0: as x \to -\infty, y \to -\infty; as x \to \infty, y \to \infty.
- Slope -0.2 < 0: as x \to -\infty, y \to \infty; as x \to \infty, y \to -\infty. The outputs are positive near 0, but past x = 1500 they are negative and keep falling.
- A parabola opening up, since 3 > 0: as x \to \pm\infty, y \to \infty. In x(3x - 40), far out on either side both factors share a sign.
- Exponential decay, base 0.9 between 0 and 1: as x \to -\infty, y \to \infty; as x \to \infty, y \to 0.
- Both ends level toward 4: as x \to \pm\infty, f(x) \to 4. On the left the outputs are just above 4 (4.02, then 4.002) and on the right just below (3.98, then 3.998).
- As t \to \infty, V(t) \to 12: the value gets closer and closer to 12 dollars. The coin keeps losing value, more and more slowly, settling toward 12 dollars without sinking below it. A negative t would be a time before Wilda bought the coin, so the inputs are t \geq 0 and the graph starts at t = 0, with no left end.
- p(10) = 32, p(100) = 9032, p(1000) = 990{,}032, p(-100) = 11{,}032 and p(-1000) = 1{,}010{,}032. As x \to \pm\infty, p(x) \to \infty, as for any parabola opening up. Her rows sit in the middle, to the left of and at the lowest point (5, 7), where the graph is still coming down; past it the graph turns and rises.
- Along y = mx + b each step of 1 changes y by m, so n steps change it by nm. With m \neq 0, the size of nm grows without bound as n grows, so the outputs cannot stay close to any one value on either side. Leveling off needs the changes to shrink, and the steps of a line are all the same size.
- The constant function y = 3 levels toward 3 at both ends and lies on the line the whole way. For one that crosses: let the far left run just under the dashed line y = 3, climb above it through the middle, then come back down and run just above it on the far right. Both ends level toward 3, and the curve crosses the line once on the way.