Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically

Learning goal: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically, no matter how large the linear or quadratic quantity starts or how slowly the exponential quantity begins. Explain why, using the way each kind of function grows from one step to the next.

Three ways to grow. A quantity increasing linearly adds the same amount each step: L(x) = 10x + 50 starts at 50 and adds 10 every time x increases by 1. A quantity increasing quadratically adds an amount that itself grows by a constant: Q(x) = 3x² adds 3, then 9, then 15, then 21, and so on, each addition 6 more than the last. A quantity increasing exponentially multiplies by the same factor each step: E(x) = 2ˣ starts at 1 and doubles every time x increases by 1, so it adds 1, then 2, then 4, then 8, each addition twice the last. The additions tell the story. The linear function's additions never change. The quadratic's additions grow, but only by adding a fixed 6 each time. The exponential's additions grow by multiplying, and multiplying beats adding in the long run.

A table that shows the overtaking. Tabulate the three functions for x from 0 to 12. At x = 0 the values are L = 50, Q = 0, E = 1; the linear function is far ahead, and the exponential is only 1, just above the quadratic's 0, which it falls behind at x = 1. At x = 5 they are L = 100, Q = 75, E = 32; the exponential is still last. At x = 6 they are L = 110, Q = 108, E = 64. At x = 7 they are L = 120, Q = 147, E = 128: the quadratic has passed the linear function, and the exponential has just passed the linear function too, since 128 > 120. At x = 8 they are L = 130, Q = 192, E = 256: the exponential has now passed the quadratic, since 256 > 192. From then on the gap only widens: at x = 10, L = 150, Q = 300, E = 1,024; at x = 12, L = 170, Q = 432, E = 4,096. Between x = 0 and x = 12 the linear function gained 120, the quadratic gained 432, and the exponential gained 4,095. Once the exponential has passed both, from x = 8 on, it stays ahead, because at every later step it adds more than the other two add (at x = 0 it is briefly ahead of the quadratic without this being true, which is why it falls behind again); the exponential's next addition is its whole current value, while the quadratic's next addition is 6x + 3 and the linear function's is 10.

A graph that shows the same thing. Plotted on the same axes, the linear function is a straight line of constant steepness. The quadratic is a curve that gets steeper at a steady rate. The exponential starts nearly flat. At x = 0 it is just above the parabola, 1 against 0, but it falls below the parabola before x = 1 (the curves cross near x = 0.75) and stays below both for a while. Then it bends upward more sharply than either, crossing the line near x = 6.9 and the parabola again near x = 7.3. On a graph with the vertical axis running to 500, the exponential leaves the top of the picture just before x = 9, since 2⁹ = 512, while the line and the parabola remain inside it all the way to x = 12. The last two crossings, after which the exponential never falls behind again, are the observation the standard names: the exponential quantity eventually exceeds both the linear quantity and the quadratic quantity.

Eventually can take a long time. The word eventually is exact. A slowly growing exponential with a small base may lag for a very long time before it overtakes, but it does overtake. Compare 1.5ˣ with 100x. At x = 10 the exponential is about 57.7 against 1,000; at x = 15 it is about 437.9 against 1,500; at x = 18 it is about 1,477.9 against 1,800, still behind; at x = 19 it is about 2,216.8 against 1,900, ahead for good (it also led very briefly at the start, since 1.5⁰ = 1 is more than 100 × 0 = 0, until just after x = 0.01); and at x = 25 it is about 25,251 against 2,500, ten times the linear value. Compare the even slower 1.1ˣ with x². At x = 50 the exponential is about 117.4 against 2,500. At x = 90 it is about 5,313 against 8,100. At x = 96 it is about 9,412 against 9,216, ahead by a whisker, and at x = 100 it is about 13,781 against 10,000. The last crossing came between x = 95 and x = 96 (the two curves also cross near x = 1, where 1.1ˣ briefly leads because 0² and 1² are so small), and the exponential never falls behind again. A base of 1.02 growing against 5x + 100 does not overtake until somewhere between x = 350 and x = 400: at x = 350 the values are about 1,023 against 1,850, and at x = 400 about 2,755 against 2,100. None of these delays contradicts the claim; they show what eventually means.

Why it must happen. Think about the ratio of consecutive values. For the exponential bˣ with b > 1 the ratio from one step to the next is always b, a fixed number larger than 1. For a linear function the ratio (10x + 60)/(10x + 50) gets closer and closer to 1 as x grows, and for a quadratic the ratio 3(x + 1)²/(3x²) = (1 + 1/x)² also gets closer and closer to 1. So there is a step beyond which the linear and quadratic quantities are each growing by a factor smaller than b every step while the exponential is growing by exactly b. From that step on the exponential gains on them by a factor each step, and a quantity that gains by a factor each step overtakes any fixed lead. That is the reasoning behind the observation; the tables and graphs are the evidence, and the ratio argument is why the evidence is not a coincidence.

The misconception. Students often judge growth by the first few values. In the table above the exponential is the smallest of the three functions for x = 1, 2, 3, 4, 5 and 6 (at x = 0 it is 1, just ahead of the quadratic's 0), and a student who stops there concludes that exponential growth is slow. It is slow at first, precisely because it multiplies a small number. The judgment must be made over a long enough interval, which is why the standard says to observe using graphs and tables rather than to evaluate at a single point. The same misconception appears in reverse when a student assumes every increasing curve is exponential; a parabola also gets steeper, but its additions grow by adding, not by multiplying, and an exponential will eventually pass it.

Worked table comparison: Compare A(x) = 20x + 100 with B(x) = 3ˣ for whole numbers x. At x = 0: 100 and 1. At x = 2: 140 and 9. At x = 4: 180 and 81. At x = 5: 200 and 243. So the exponential first exceeds the linear function at x = 5, since 3⁵ = 243 > 200 while 3⁴ = 81 < 180. The additions explain it: A adds 20 every step, B triples every step, and once B is adding more than 20 per step, which happens as soon as B exceeds 10, its lead can only grow.

Worked graph comparison with a quadratic: Compare Q(x) = x² + 20 with E(x) = 2ˣ from the graphs, by computing the plotted points. At x = 2: 24 and 4. At x = 4: 36 and 16. At x = 5: 45 and 32. At x = 6: 56 and 64. The exponential crosses above the parabola between x = 5 and x = 6, and at x = 10 the values are 120 and 1,024. On the graph the parabola is above the exponential curve on the left, the two curves cross once between 5 and 6, and the exponential is steeper everywhere to the right of the crossing.

Worked slow overtaking: Compare C(x) = 50x with D(x) = 1.5ˣ. At x = 18, C = 900 and D ≈ 1,477.9: already ahead. At x = 15, C = 750 and D ≈ 437.9: behind. At x = 17, C = 850 and D ≈ 985.3: ahead. At x = 16, C = 800 and D ≈ 656.8: behind. So the final crossing lies between x = 16 and x = 17 (D is also briefly ahead at the very start, 1 against 0 at x = 0, until a first crossing near x = 0.02). This exponential grew by only 50 percent per step and started at 1, yet it overtook a line rising 50 units per step within 17 steps, and at x = 25 the comparison is 1,250 against about 25,251.

Representation task: Build a table for x = 0 to 12 with columns for L(x) = 10x + 50, Q(x) = 3x² and E(x) = 2ˣ, and add three more columns giving the amount each function adds from one row to the next. Circle the row from which on E stays above L and the row from which on E stays above Q. Then graph the three functions on axes running to x = 12 and y = 500, mark all three crossings (E meets Q near x = 0.75 and again near x = 7.3, and meets L near x = 6.9), and write two sentences: one describing how each function's additions change down the table, and one explaining why the exponential's lead cannot be lost once it is gained.

Independent performance: Choose your own linear function with a large starting value, your own quadratic, and an exponential with a base between 1.2 and 2 and a starting value of 1. Predict, before computing, roughly where the exponential will overtake each of the others. Then build a table wide enough to find the crossing after which the exponential stays ahead of each of the others, bracketing each between two consecutive whole-number inputs (note any earlier crossing near x = 0 as well), and graph the three functions over that interval using technology or by hand. Finally write an argument, using the ratio of consecutive values, for why the exponential had to overtake both eventually, and compare your prediction with the crossings you found. Success criteria: each crossing is bracketed by a pair of consecutive inputs with the values on both sides shown; the graph shows the exponential below the others at first and above them after the crossings; and the argument mentions that the exponential multiplies by a fixed factor each step while the others' consecutive ratios approach 1.