HSF.LE.A.3: Exponential Growth Eventually Exceeds Polynomial Growth
In plain English: HSF.LE.A.3 is the Common Core functions standard that asks students to observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically or as any polynomial function. Repeated multiplication by a factor greater than 1 always outruns polynomial growth in the long run, even from far behind. It is usually taught in Algebra I.
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Common Core State Standards for Mathematics · Domain: Linear, Quadratic, and Exponential Models (LE) · Cluster: Construct and compare linear, quadratic, and exponential models and solve problems Also written as HSF-LE.A.3 or F-LE.3 · Official standard
Students use tables and graphs to observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically or as any polynomial. They compare pairs of functions such as 100x and 2x, x² and 2x, and x³ and 2x, and find the input after which the exponential stays ahead.
The lesson stresses the word "eventually." Many exponentials start behind, and small graphing windows can hide the crossing completely. Students learn to extend tables in larger steps, adjust windows, and explain the pattern with ratios: an exponential multiplies by the same factor every step, while a polynomial's step-to-step ratio shrinks toward 1.
Learning Objectives
By the end of this lesson, students will be able to:
Use a table to find the input after which an exponential function stays greater than a linear function
Use tables and graphs to show that an exponential function eventually exceeds a quadratic function, even when the quadratic is ahead at first
Show with tables and graphs that exponential growth eventually exceeds any polynomial, including cubic and higher-degree ones
Choose a graphing window or table step that makes the long-run behavior visible, and explain why a small window can mislead
Prior Knowledge Required
Students should already be comfortable with:
Evaluating expressions with whole-number exponents 8.EE.A.1
Telling linear and exponential growth apart by differences and factors HSF.LE.A.1
Graphing linear, quadratic and exponential functions HSF.IF.C.7
Reading and building tables of values, by hand or with a spreadsheet
Pose the choice below and take a quick vote before anyone computes.
Warm-Up Prompt
"You can be paid for a 30-day job in one of two ways. Offer 1: $500 every day. Offer 2: 1 cent on day 1, 2 cents on day 2, 4 cents on day 3, doubling each day. On which day does Offer 2 first pay more than $500 for that day? Which offer pays more on day 30?"
Let pairs build a table with a calculator. On day n, Offer 2 pays 0.01 · 2n - 1 dollars: $327.68 on day 16 and $655.36 on day 17, so from day 17 on each daily payment beats $500. On day 30 it pays $5,368,709.12. Ask how many students voted for Offer 1 and why it looked better: the doubling starts tiny and stays small for over two weeks. The lesson question is whether this always happens.
Direct Instruction20 minutes
What "eventually exceeds" means. A quantity g eventually exceeds a quantity f if there is some input after which g(x) > f(x) and stays greater. The exponential may start above, fall behind for a while and then pass for good, as in Example 2, so "eventually" is about the long run, not the first few rows. Show how to find the passing point with a table:
Build a table of both functions for whole-number inputs, starting at 0.
Extend it in bigger steps (x = 10, 20, 30, ...) until the exponential is clearly ahead, then fill in the whole numbers around the change.
Find the last input where the other function is still ahead or tied; the exponential leads from the next input on.
Confirm with a graph, adjusting the window so both curves and the crossing are visible.
Explain why the lead lasts: each step multiplies the exponential by the same factor b > 1, while the step-to-step ratio of a linear or polynomial function, such as (x + 1)² ÷ x², shrinks toward 1. Once the exponential is ahead and its ratio is larger, the gap keeps growing.
Use Diagram 1 to compare y = 6x, y = x² and y = 2x on one graph: at x = 5 the exponential passes both. Then work through the examples. Emphasize the standard's words "increasing linearly, quadratically, or (more generally) as a polynomial function": the same outcome holds for every polynomial, even one with a large leading coefficient or a high degree.
Linear versus exponential
Compare f(x) = 100x and g(x) = 2ˣ in a table for whole numbers x.
Equation: x = 9: 900 versus 512; x = 10: 1,000 versus 1,024; g(x) > f(x) for every whole number x ≥ 10
Quadratic versus exponential
Compare q(x) = x² and g(x) = 2ˣ for x = 0 to 6.
Equation: Equal at x = 2 and x = 4, x² ahead only at x = 3, and 2ˣ > x² for every whole number x ≥ 5
Polynomial (cubic) versus exponential
Compare p(x) = x³ and g(x) = 2ˣ, graphed in Diagram 2.
Equation: x = 9: 729 versus 512; x = 10: 1,000 versus 1,024; 2ˣ > x³ for every x ≥ 10
Linear versus exponential in context
A tutoring website has 1,000 users. Plan L adds 400 users per month; Plan E grows 8% per month.
Equation: Month 35: 15,000 versus about 14,785; month 36: 15,400 versus about 15,968; Plan E leads from month 36 on
A slow exponential still wins
Compare y = x with y = 1.01ˣ, which grows only 1% per unit.
Equation: 1.01ˣ falls behind just after x = 1, is still behind at x = 651 and leads for good from x = 652
Use Diagram 2 for Example 3. For Example 5, show a spreadsheet: at x = 100, 1.01x is only about 2.7, but by x = 1,000 it is about 20,959. The base only has to be greater than 1.
Guided Practice15 minutes
Pairs use graphing technology and a table for each comparison. They record the window that shows the crossing and the first whole number after which the exponential stays ahead.
3x versus 20x²: at x = 5, 243 versus 500; at x = 6, 729 versus 720. The exponential leads from x = 6 on. A window of 0 ≤ x ≤ 8 and 0 ≤ y ≤ 1,500 shows it
1.5x versus 10x + 100: at x = 13, about 194.6 versus 230; at x = 14, about 291.9 versus 240. The exponential leads from x = 14 on. The standard window of 10 by 10 shows nothing useful here; students need about 0 ≤ x ≤ 20 and 0 ≤ y ≤ 400
Ask each pair: "What would you tell someone who graphed only 0 ≤ x ≤ 10?"
Independent Practice15 minutes
Students build tables on their own (a spreadsheet is fine) and find the first whole number x after which the exponential stays ahead.
3x versus 100x (from x = 6: 729 versus 600; at x = 5, 243 versus 500)
1.3x versus x² (from x = 25: about 705.6 versus 625; at x = 24, about 542.8 versus 576)
4x versus x4 (tied at x = 2 and x = 4, x4 ahead at x = 3, and 4x ahead from x = 5: 1,024 versus 625)
1.5x versus x³ (from x = 24: about 16,834 versus 13,824; at x = 23, about 11,223 versus 12,167)
For each, students write one sentence about what a graph in a small window would have suggested.
Closure5-10 minutes
Exit ticket: (1) After which whole number does 2x pass 12x and stay above it? (x = 7: 128 versus 84; at x = 6, 64 versus 72.) (2) True or false: y = 1,000,000x stays ahead of y = 1.001x forever. Explain using the idea of this lesson. (False: any exponential with base greater than 1 eventually passes any linear function, even if the crossing is far out.)
Differentiation Strategies
For Struggling Students
Provide partially filled tables with the rows around the crossing already chosen, so students focus on comparing values
Use a spreadsheet with one column per function and a third column that shows "exponential ahead?" as yes or no
Start with base 2 comparisons, where the values are whole numbers, before bases such as 1.3 or 1.08
For Advanced Students
Ask students to compute the ratio (x + 1)³ ÷ x³ at x = 10, 50 and 100 and explain how it compares with the base 2 or 1.5 of an exponential
Ask students to find a base b > 1 for which bx does not pass x² until after x = 1,000, and test it with a spreadsheet
Ask students to compare 2x with x8 and find where the crossing happens (the exponential leads for good from x = 44)
Assessment Guidance
What to Look For
Check that students give a specific input after which the exponential stays ahead and support it with table values on both sides of that input. Listen for the word "eventually" used correctly: a student who says "the exponential is always bigger" or "the quadratic is always bigger" based on a short table has not yet understood the standard. Look for sensible windows and table steps, and for an explanation that refers to repeated multiplication versus repeated addition.
02
Classroom Activities
3 Activities
1
Race Tables
20 minPairs
Each pair gets 3 race cards. Each card names an exponential "runner" and a polynomial "runner." Pairs build a table, predict the winner at x = 30 and find where the exponential takes the lead for good.
The 3 Race Cards
Card 1: 2x versus 5x². At x = 8: 256 versus 320. At x = 9: 512 versus 405. The exponential leads from x = 9
Card 2: 1.1x versus 10x. At x = 50: about 117 versus 500. The exponential leads from x = 69 (about 718 versus 690)
Card 3: 2x versus x4. Tied at x = 16 (65,536 each); the exponential leads from x = 17
Procedure
Before computing, each partner predicts which runner leads at x = 30 and writes the prediction down
Pairs build the table by hand for small x and with a spreadsheet for large x, using bigger steps first and then filling in whole numbers
Pairs circle the last row where the polynomial is ahead or tied and the first row of the lasting exponential lead
Discussion Questions
Card 3 has a tie at x = 16. How can you be sure the exponential stays ahead after it?
Card 2 has a slow base, 1.1. Did the exponential still win? What changed compared with Card 1?
Which prediction was furthest off, and why?
2
Window Detective
20 minPairs
Pairs graph two comparisons with technology in the standard window, where the polynomial seems to win, and then change the window until the exponential passes it. The activity shows why "eventually" needs large inputs.
Comparisons
y = 1.1x and y = x². The exponential leads for good from x = 96
y = 1.2x and y = 50x². The exponential leads for good from x = 68
Procedure
Graph both functions in a window of -10 ≤ x ≤ 10 and -10 ≤ y ≤ 10 and write what the graph seems to show
Change Xmax and Ymax step by step and record each window until the crossing is visible
Use the table feature to find the whole number where the exponential takes the lasting lead, and check it against the graph
Modification for Distance Learning
Share a graphing app link with both functions already entered. Students post screenshots of their "misleading" window and their "revealing" window with a one-sentence caption.
3
Rice on the Chessboard
15 minWhole class, then pairs
Retell the old story: a ruler places 1 grain of rice on the first square of a chessboard, 2 on the second, 4 on the third, doubling each time. Pairs compare this plan with two polynomial plans.
Procedure
Square n gets 2n - 1 grains under the doubling plan. Plan B puts 1,000n grains on square n. Plan C puts n³ grains on square n
Pairs build a table and find where doubling passes each plan for good: Plan B from square 15 (16,384 versus 15,000), Plan C from square 12 (2,048 versus 1,728)
Pairs compute the grains on the last square, 263, which is about 9.2 quintillion, and compare it with Plan B's 64,000
Challenge Variation
Pairs find the total grains on all 64 squares under the doubling plan, 264 - 1, by noticing that each partial total is one less than the next square. Ask them to estimate how that total compares with the world's yearly rice harvest using a published figure they look up.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Linear, Quadratic and Exponential Growth on One Graph
The three functions are drawn to scale for 0 ≤ x ≤ 6. At x = 4 the exponential is behind the line (16 versus 24) and tied with the parabola (16 versus 16). At x = 5 it passes both (32 versus 30 and 25), and by x = 6 it is almost double both of them. The table gives the exact values.
Diagram 2: The Exponential y = 2ˣ Overtakes the Cubic y = x³
Graphs of y = x³ and y = 2x drawn to scale for 0 ≤ x ≤ 12. The cubic is ahead from about x = 1.37 to about x = 9.94, which a window that stops at x = 9 would suggest is permanent. At x = 10 the values are 1,000 and 1,024, and by x = 12 the exponential is more than twice the cubic.
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Homework Assignment
~30 min
HSF.LE.A.3 Homework: When Does the Exponential Win?
Directions: Use tables, and a graphing tool where asked. For each comparison, give the table rows on both sides of the crossing and state the first whole number after which the exponential stays ahead. You may use a spreadsheet for large inputs.
Part 1: Linear and Quadratic (Problems 1-3)
Make a table of f(x) = 40x and g(x) = 2x for x = 0 to 10. After which whole number does g stay ahead of f? At which inputs is g ahead before that?
Make a table of q(x) = 3x² and g(x) = 1.6x for x = 0, 5, 10, 12, 13, 14 and 15, rounding to one decimal place. Find the first whole number after which g stays ahead. How would a graph with 0 ≤ x ≤ 10 mislead you?
A new app has 5,000 users. Model A adds 800 users per month. Model B grows the user count by 10% per month. Make a table for months 0, 5, 9, 10, 11 and 12, rounding to whole users. When does Model B first have more users, and what happens in month 10?
Part 2: Polynomials in General (Problems 4-6)
Compare p(x) = x4 and g(x) = 3x in a table for x = 0 to 9. Describe every change in which function is ahead, and give the first whole number after which g stays ahead.
Graph y = x5 and y = 2x with graphing technology. Record a window in which the polynomial looks larger everywhere, then a window that shows the crossing. Use the table feature to find the first whole number after which 2x stays ahead.
A classmate claims that y = 1,000x² will always be larger than y = 1.05x because 1,000 is so large and 1.05 is so close to 1. Test the claim with a table at x = 100, 200, 300 and 400. Was the classmate right? Explain what the table shows.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Tables
Accurate values on both sides of every crossing
Minor rounding or arithmetic errors
Missing or mostly incorrect
Crossing Point
Correct first whole number of the lasting lead, supported by the table
Crossing located but not exact
No crossing identified
Graphs and Windows
Misleading and revealing windows both recorded and explained
One window or no explanation
No graph work
Explanation
Uses "eventually" correctly and connects to repeated multiplication
Correct conclusion, weak reasoning
Incorrect conclusion
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Quiz: 20 Questions
Interactive, with answers
Instructions
Several questions need a calculator. Pick or write an answer, then reveal the explanation. The score updates as you work, and Reset quiz clears all answers.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Let f(x) = 500x and g(x) = 1.02x. Which statement is true?
Answer: A
Any exponential with a base greater than 1 eventually exceeds any linear function, even one with a slope of 500; here the crossing is far out, but it happens. Choice B is what a small table suggests. Choice D is false: the base only has to be greater than 1.
Question 2 of 20 · Multiple Choice
A table compares 2x and 10x: x = 4 gives 16 and 40; x = 5 gives 32 and 50; x = 6 gives 64 and 60; x = 7 gives 128 and 70. From which whole number on does 2x stay ahead?
Answer: C
At x = 5, 32 < 50, but at x = 6, 64 > 60, and the gap keeps growing (128 versus 70 at x = 7). Choice B picks the last row where 10x is still ahead. Choice D ignores the row x = 6.
Question 3 of 20 · Multiple Choice
For which whole numbers x ≥ 0 is 2x² greater than 2x?
Answer: A
The values are x = 0: 0 and 1; x = 1: 2 and 2; x = 2: 8 and 4; x = 3: 18 and 8; x = 4: 32 and 16; x = 5: 50 and 32; x = 6: 72 and 64; x = 7: 98 and 128. So 2x² is greater only for x = 2 through 6, and from x = 7 on 2x stays ahead. Choice B counts the tie at x = 1 as "greater." Choice C reverses the long-run result. Choice D ignores the rows x = 2 to 6.
Question 4 of 20 · Multiple Choice
Which function eventually exceeds all the others?
Answer: D
The exponential y = 1.005x has a base greater than 1, so it eventually exceeds every linear, quadratic and higher-degree polynomial function, however large their coefficients. Choice C is the fastest of the polynomials, but a polynomial never outgrows an exponential.
Question 5 of 20 · Multiple Choice
A table shows f(x) = x³ and g(x) = 3x: x = 1 gives 1 and 3; x = 2 gives 8 and 9; x = 3 gives 27 and 27; x = 4 gives 64 and 81. Which conclusion does the table support?
Answer: B
The table shows a tie at x = 3 and 3x ahead at x = 4 (81 versus 64); each step multiplies 3x by 3, while (x + 1)³ ÷ x³ is already less than 3, so the lead grows. Choice C goes beyond the table: between x = 2 and x = 3 there is a short interval, near x = 2.5, where x³ is slightly larger.
Question 6 of 20 · Multiple Choice
Why does an exponential function with base greater than 1 eventually pass a linear function?
Answer: A
The exponential's step-to-step change is a fixed percent of a growing quantity, so it grows without limit, while the line adds the same amount every step. Choices B and C are false in general: 2x starts below 100x at x = 1 and still passes it. Choice D confuses an increasing line with a decreasing one.
Question 7 of 20 · Multiple Choice
A student graphs y = 1.05x and y = 20x in the window 0 ≤ x ≤ 50 and concludes that 20x is always larger. What is wrong with the conclusion?
Answer: C
At x = 50, 1.05x is only about 11.5 while 20x is 1,000, but the exponential keeps multiplying by 1.05 and passes 20x for good at x = 167. A table in larger steps or a wider window reveals this. Choice A is the error the student made.
Question 8 of 20 · Multiple Choice
At x = 13, which of these is greatest?
Answer: D
At x = 13: 500x = 6,500, x³ = 2,197, 40x² = 6,760 and 213 = 8,192, so the exponential is greatest. One step earlier, at x = 12, both 500x = 6,000 and 40x² = 5,760 were still ahead of 212 = 4,096. A student who picks A or C may assume that a large coefficient keeps the polynomial ahead; choice B is the smallest of the four at x = 13.
Question 9 of 20 · Multiple Choice
A company has 10,000 customers. Plan L adds 2,000 customers per year. Plan E grows the customer count by 10% per year. After how many whole years does Plan E first have more customers?
Answer: D
At t = 14: Plan L has 38,000 and Plan E about 37,975, still slightly fewer. At t = 15: 40,000 versus about 41,772. Plan E leads from year 15 on. Choice C misses by a small margin, so it is worth computing carefully rather than estimating.
Question 10 of 20 · Multiple Choice
A table shows q(x) = x² and e(x) = 1.5x at x = 2, 5, 10 and 15: q = 4, 25, 100, 225 and e ≈ 2.25, 7.59, 57.67, 437.89. Which statement is correct?
Answer: C
At x = 10, q is ahead (100 versus 57.67); at x = 15, e is ahead (437.89 versus 225). So e passes q between 10 and 15; filling in the rows shows the lasting lead starts at x = 13. Choice A is the error of stopping the table at x = 10. Choice B is wrong: q is still ahead at x = 5.
Question 11 of 20 · Multiple Choice
Compare p(x) = x10 and g(x) = 2x. Which statement is true?
Answer: A
At x = 20, p is about 1013 and g is about 106, but the exponential still passes the polynomial; a spreadsheet shows the lasting lead starts at x = 59. Choice C confuses the exponent with a crossing point: 1010 is much larger than 210 = 1,024.
Question 12 of 20 · Multiple Choice
The step-to-step increase of 2x is 2x + 1 - 2x = 2x, while 50x always increases by 50. From which whole number x is the increase of 2x first larger than 50?
Answer: B
The increase from x to x + 1 is 2x: 32 at x = 5 and 64 at x = 6. So from x = 6 on, the exponential gains more than 50 per step, and its gains keep doubling. Choice A gives an increase of only 32. Choice C divides 50 by 2 as if the exponential were linear.
Question 13 of 20 · Multiple Choice
A student makes a table of y = x² and y = 1.2x for x = 0 to 20. At x = 20, x² = 400 and 1.2x ≈ 38.3, so the student says x² stays ahead. What is the best response?
Answer: B
At x = 40 the values are 1,600 and about 1,470, and at x = 41 they are 1,681 and about 1,764, so the exponential leads from x = 41 on. Any base greater than 1 works. Choice C mistakes a slow base for no growth. Choice D ignores the table, where x² is clearly ahead at x = 20.
Question 14 of 20 · Multiple Choice
For large values of x, which order is correct from smallest to largest?
Answer: D
For large x, a linear function is exceeded by a cubic, and every polynomial is eventually exceeded by an exponential with base greater than 1. Choice A is the order for moderately small x, such as x = 5. Choice C puts the exponential below the cubic, which is only true before the exponential catches up.
Question 15 of 20 · Short Answer
Complete a table for f(x) = 25x and g(x) = 2x at x = 5, 6, 7 and 8. From which whole number on does g stay ahead of f?
x = 5: 125 and 32; x = 6: 150 and 64; x = 7: 175 and 128; x = 8: 200 and 256. From x = 8 on, g stays ahead, because g doubles each step while f adds only 25.
Question 16 of 20 · Short Answer
Compare q(x) = 4x² and g(x) = 2x at x = 7, 8, 9 and 10. What do you notice at x = 8, and what happens after it?
x = 7: 196 and 128; x = 8: 256 and 256; x = 9: 324 and 512; x = 10: 400 and 1,024. The functions tie at x = 8, and from x = 9 on the exponential is ahead, with a gap that keeps growing.
Question 17 of 20 · Short Answer
A nursery has 2,000 seedlings. Plan L adds 300 seedlings per year. Plan E increases the number of seedlings by 8% per year. Use a table to find when Plan E first has more seedlings.
At t = 15: Plan L has 6,500 and Plan E about 6,344. At t = 16: 6,800 versus about 6,852. Plan E has more from year 16 on. Plan L is ahead for the first 15 years, which a short table would suggest is permanent.
Question 18 of 20 · Short Answer
Let p(x) = x³ + 100x and g(x) = 2x. Evaluate both at x = 10, 11 and 12. What do you conclude?
x = 10: 2,000 and 1,024; x = 11: 2,431 and 2,048; x = 12: 2,928 and 4,096. The exponential passes the polynomial between x = 11 and x = 12 and stays ahead from x = 12 on. Adding 100x to x³ only delays the crossing by a couple of steps.
Question 19 of 20 · Short Answer
Explain, using ratios of consecutive outputs, why 1.05x eventually passes x4. Use the ratio (x + 1)4 ÷ x4 at x = 100 in your answer.
Each step multiplies 1.05x by 1.05. For x4, the step ratio is (x + 1)4 ÷ x4 = (1.01)4, which at x = 100 is about 1.0406 and keeps getting closer to 1. Once the exponential's ratio (1.05) is bigger than the polynomial's, the exponential grows by a bigger percent every step, so it closes any gap and then pulls ahead for good.
Question 20 of 20 · Short Answer
Using a table or graphing technology, find the first whole number after which 1.5x stays greater than 50x. Give the table values on both sides.
x = 16: 1.516 ≈ 656.8 and 50x = 800; x = 17: 1.517 ≈ 985.3 and 50x = 850. From x = 17 on, 1.5x stays ahead.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does HSF.LE.A.3 mean?
HSF.LE.A.3 means students can show, with graphs and tables, that exponential growth eventually beats linear, quadratic and other polynomial growth. For example, 2x is behind 100x until x = 10 and ahead for every whole number after that. The standard asks students to observe this pattern, not to prove it formally.
Is HSF.LE.A.3 taught in Algebra 1 or Algebra 2?
It is usually taught in Algebra I, once students can graph linear, quadratic and exponential functions. Algebra II and precalculus return to it with higher-degree polynomials and logarithms, and calculus explains it with limits.
What does "eventually exceeds" mean?
It means there is an input after which the exponential is greater and stays greater. The exponential may be ahead at first, fall behind, and then pass for good: 2x is ahead of x² at x = 0 and 1, tied at 2, behind at 3, tied at 4 and ahead for every whole number from 5 on.
Does an exponential beat a polynomial even if the polynomial has a huge coefficient?
Yes. A large coefficient or a high degree only moves the crossing further out. For example, 1.1x is far behind 10,000x³ at x = 100 (about 13,781 versus 10 billion), but it stays ahead from x = 274 on. The only condition is that the exponential is increasing: base greater than 1 and a positive coefficient.
Why can a graph make the polynomial look like it wins?
Because the crossing can lie outside the window. With 0 ≤ x ≤ 50, the graph of 1.04x stays near the x-axis while 30x reaches 1,500. Widening the window, or building a table in larger steps, shows the exponential passing for good at x = 225. Teaching students to change Xmax and Ymax is part of this standard.
How can students explain why exponential growth always wins?
Compare step-to-step ratios. An exponential multiplies by the same factor b every step. A polynomial's step ratio, such as (x + 1)² ÷ x², gets closer and closer to 1 as x grows. Once the polynomial's ratio is below b, the exponential grows by a larger percent every step and eventually overtakes it. This is an informal argument that fits the "observe" level of the standard.
What are common mistakes with HSF.LE.A.3?
Common mistakes are stopping a table too early and concluding that the polynomial always wins, reading a tie as a lasting lead, using a graphing window that is too small, and thinking a base such as 1.05 is "too small" to win. Another is claiming the exponential is ahead for all x when it is ahead only after some point.
Does this also work for exponential decay?
No. The standard is about a quantity increasing exponentially, which needs a base greater than 1. A decaying exponential such as 100(0.9)x shrinks toward 0, so an increasing linear or polynomial function eventually exceeds it instead.
What real situations show this idea?
Comparisons of a fixed yearly increase with a percent increase are common examples: savings with a fixed deposit versus compound interest, or a user base that gains a fixed number each month versus one that grows by a percent. In each case the percent plan starts slower and eventually leads. Doubling stories such as rice on a chessboard make the size of the gap vivid.
How does HSF.LE.A.3 connect to other standards?
It builds on telling linear and exponential growth apart (HSF.LE.A.1) and on graphing functions (HSF.IF.C.7). It sits next to comparing functions given in different forms (HSF.IF.C.9), and it prepares students to solve exponential equations with logarithms (HSF.LE.A.4), which is how the exact crossing points are found later.
07
Related Standards
6 standards
These standards connect to HSF.LE.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.LE.A.1Prerequisite
Distinguish situations modeled by linear functions from exponential ones
