Linear vs. Exponential Functions

By

To compare linear vs. exponential functions, look at how the output changes each step. A linear function adds the same amount each time, so its graph is a straight line. An exponential function multiplies by the same number each time, so its graph curves. In the long run, exponential growth always wins.

Before this lesson: You should know slope-intercept form, exponential functions, and average rate of change.

What Is the Difference Between Linear and Exponential Functions?

Two new video channels start at the same time. Channel A has 1,000 followers and gains 400 each month. Channel B has only 100 followers, but its count doubles each month.

At first, Channel A is far ahead. But doubling gets bigger every month, while adding 400 stays the same. By month 5, Channel B has 3,200 followers and Channel A has 3,000. After that, B pulls away fast.

Channel A is a linear function. Channel B is an exponential function. The key question is always the same: does the amount add the same number, or multiply by the same number?

Graph of followers against months from 0 to 6. Channel A, a solid blue straight line, starts at 1,000 and adds 400 each month: 1,000, 1,400, 1,800, 2,200, 2,600, 3,000, 3,400. Channel B, a dashed orange curve, starts at 100 and doubles each month: 100, 200, 400, 800, 1,600, 3,200, 6,400. The curve crosses the line between month 4 and month 5.
Channel A adds 400 a month and makes a line. Channel B doubles and makes a curve that passes A between months 4 and 5.

Linear vs. Exponential Functions: The Rules

Linear functionExponential function
Equationy = mx + by = a \cdot b^x
Each step in xAdds m to yMultiplies y by b
In a tableSame differenceSame ratio
GraphStraight lineCurve
Rate of changeStays the sameKeeps getting bigger (or smaller)
Words to look for“adds 5 each,” “$20 per hour”“doubles,” “halves,” “grows 5% a year”

The difference is what you get when you subtract one output from the next. The ratio is what you get when you divide one output by the one before it.

Two rows of boxes. Top row, linear: 4, 7, 10, 13, with an arrow labeled plus 3 between each pair. Bottom row, exponential: 3, 6, 12, 24, with an arrow labeled times 2 between each pair.
Linear outputs go up by the same amount. Exponential outputs go up by the same factor.

How to Tell Them Apart in a Table

  1. Check that the x-values go up by the same amount each time.
  2. Subtract each output from the next one. If every difference is the same, the function is linear.
  3. If not, divide each output by the one before it. If every ratio is the same, the function is exponential.
  4. If neither one stays the same, the function is neither linear nor exponential.
  5. Write the equation. Linear: m is the difference and b is the value at x = 0. Exponential: b is the ratio and a is the value at x = 0.

An exponential function that grows will always pass a linear function that grows, sooner or later. A bigger start only delays it.

Worked Examples

Example 1: Linear, exponential or neither?

Decide whether each table is linear, exponential or neither. Write an equation if you can.

x0123
Table A: y471013
Table B: y361224
Table C: y12510

Table A: The differences are 3, 3, 3. They match, so it is linear. The slope is 3 and the start is 4, so y = 3x + 4.

Table B: The differences are 3, 6, 12. They don’t match. The ratios are 6 \div 3 = 2, 12 \div 6 = 2 and 24 \div 12 = 2. They match, so it is exponential: y = 3 \cdot 2^x.

Table C: The differences are 1, 3, 5. The ratios are 2, 2.5 and 2. Neither one stays the same, so it is neither.

Example 2: Linear or exponential from words

Decide whether each situation is linear or exponential. Then write a function.

  1. A gym costs $25 to join plus $15 per month. Linear, because it adds $15 each month: C = 15m + 25.
  2. A town of 8,000 people grows 5% per year. Exponential, because it multiplies by 1.05 each year: P = 8000(1.05)^t.
  3. A 20 cm candle burns 2 cm per hour. Linear, because it subtracts 2 each hour: h = 20 - 2t.
  4. A bacteria colony of 50 cells triples every hour. Exponential, because it multiplies by 3: N = 50 \cdot 3^t.

“Per month” with a dollar amount means add. “Percent” or “triples” means multiply.

Example 3: Write both functions through two points

Write a linear function and an exponential function that both pass through (0, 6) and (2, 54).

Linear: The slope is \frac{54 - 6}{2 - 0} = \frac{48}{2} = 24. The start is 6. So y = 24x + 6.

Exponential: The start is a = 6. From x = 0 to x = 2, you multiply by b twice. So 6 \cdot b^2 = 54.

\begin{aligned} 6b^2 &= 54 \\ b^2 &= 9 \\ b &= 3 \end{aligned}

So y = 6 \cdot 3^x. Check: 6 \cdot 3^2 = 6 \cdot 9 = 54. The base of an exponential function must be positive, so b = 3, not −3.

Example 4: When does the exponential function pass the line?

Compare f(x) = 50x + 100 and g(x) = 2^x. For which whole numbers is g(x) bigger?

x02468910
f(x)100200300400500550600
g(x)1416642565121024

The line starts far ahead. At x = 9, f(9) = 550 and g(9) = 512, so the line is still ahead. At x = 10, f(10) = 600 and g(10) = 1024.

So g(x) is bigger for every whole number x \ge 10. After that, g(x) doubles each step, but f(x) only adds 50.

Example 5: Word problem with two channels

Channel A has 1,000 followers and gains 400 per month. Channel B has 100 followers and doubles each month. Write a function for each. After how many whole months does B have more followers?

Channel A adds 400, so it is linear: A(m) = 400m + 1000. Channel B doubles, so it is exponential: B(m) = 100 \cdot 2^m.

Month m3456
A(m)2,2002,6003,0003,400
B(m)8001,6003,2006,400

At month 4, A is still ahead, 2,600 to 1,600. At month 5, B is ahead, 3,200 to 3,000. So Channel B has more followers after 5 months.

Common Mistakes

MistakeExample of the errorFix
Checking differences when the x-values skipx: 0, 1, 3, 4 with y: 2, 4, 8, 10, so “not linear”First check that x goes up by the same step. Here the slope is 2 every time, so it is linear.
Calling every curve exponential1, 4, 9, 16 is “exponential”Check the ratios: 4, 2.25, 1.78. They change, so it is not exponential.
Mixing up “5” and “5%”“Grows 5% a year” written as y = 100 + 5tA percent means multiply: y = 100(1.05)^t.
Thinking a bigger start always wins“The line starts higher, so it stays higher.”A growing exponential function always passes a growing line. Make a table to see when.
Using the ratio as the slope3, 6, 12, 24 written as y = 2x + 3A same ratio means exponential: y = 3 \cdot 2^x.

Comparing Decay: Lines vs. Curves

Decreasing functions follow the same rules. A linear function that goes down subtracts the same amount each step. It will reach zero and then go negative.

An exponential function that decays multiplies by a number between 0 and 1, like \frac{1}{2}. It gets closer and closer to zero but never reaches it. You learned this in exponential growth and decay.

Time01234
Linear: subtract 20806040200
Exponential: half each step804020105

Sequences work the same way. An arithmetic sequence is linear, and a geometric sequence is exponential.

Practice Problems

  1. Is this table linear, exponential or neither? x: 0, 1, 2, 3 and y: 2, 8, 32, 128. Write an equation if you can.
  2. Is this table linear, exponential or neither? x: 0, 1, 2, 3 and y: 10, 7, 4, 1.
  3. Is this table linear, exponential or neither? x: 1, 2, 3, 4 and y: 1, 4, 9, 16.
  4. Is this table linear, exponential or neither? x: 0, 1, 2, 3 and y: 81, 27, 9, 3.
  5. A phone loses 15% of its value each year. Is its value linear or exponential?
  6. A pool fills at 40 gallons per minute. Is the amount of water linear or exponential?
  7. Write a linear function and an exponential function through (0, 4) and (1, 12).
  8. Write a linear function and an exponential function through (0, 2) and (3, 16).
  9. Compare f(x) = 20x and g(x) = 3^x. What is the first whole number x \gt 0 where g(x) \gt f(x)?
  10. Maya has $60 and saves $30 more each week. Leo has $5, and his money doubles each week. After how many whole weeks does Leo have more?
Show answers
  1. Exponential. The ratios are all 4, so y = 2 \cdot 4^x.
  2. Linear. The differences are all −3, so y = -3x + 10.
  3. Neither. The differences are 3, 5, 7 and the ratios are 4, 2.25 and about 1.78. Neither stays the same.
  4. Exponential decay. The ratios are all \frac{1}{3}, so y = 81\left(\frac{1}{3}\right)^x.
  5. Exponential. Losing 15% means multiplying by 0.85 each year.
  6. Linear. It adds 40 gallons each minute.
  7. Linear: slope 12 - 4 = 8, so y = 8x + 4. Exponential: b = 12 \div 4 = 3, so y = 4 \cdot 3^x.
  8. Linear: slope \frac{16 - 2}{3} = \frac{14}{3}, so y = \frac{14}{3}x + 2. Exponential: 2b^3 = 16, so b^3 = 8 and b = 2. y = 2 \cdot 2^x.
  9. x = 4. At x = 3: 60 vs. 27. At x = 4: f(4) = 80 and g(4) = 81.
  10. Maya: M = 30w + 60. Leo: L = 5 \cdot 2^w. Week 5: $210 vs. $160. Week 6: $240 vs. $320. Leo has more after 6 weeks.

FAQ

How do you tell if a table is linear or exponential?

Find the differences between outputs. If they are all the same, it is linear. If not, find the ratios. If the ratios are all the same, it is exponential.

Which grows faster, linear or exponential?

Exponential growth is faster in the long run. A line adds the same amount forever, but an exponential function multiplies, so its jumps keep getting bigger.

Is y = x² an exponential function?

No. In y = x^2 the variable is in the base, not the exponent. It is a quadratic function. In an exponential function like y = 2^x, the variable is the exponent.

What is the difference between linear and exponential decay?

Linear decay subtracts the same amount each step and reaches zero. Exponential decay multiplies by the same fraction each step and never quite reaches zero.

Summary

  • Linear functions add the same amount each step. Exponential functions multiply by the same number each step.
  • In a table, a same difference means linear and a same ratio means exponential.
  • Linear graphs are straight lines. Exponential graphs curve.
  • Words like “per hour” mean linear. Words like “doubles” or “5% a year” mean exponential.
  • A growing exponential function always passes a growing linear function in the end.

Previous lesson: Geometric Sequences

Next lesson: Polynomials

Back to the Algebra 1 Index

Similar Posts