Lecture: Exponential Functions
Lecture script · about 15–20 minutes spoken · Week 13
Opening
A saver compares two bank offers. A nurse tracks a dose that drops by half on a schedule. A city planner projects how big a town will be in ten years. A shopper wonders what a phone will be worth in two years. Each of them is doing the same thing: taking a starting amount and multiplying it by the same factor again and again. That pattern has a name, the exponential function, and it's the subject of this lecture.
Here's where we are in this course. In Week 1 you learned the exponent rules, including the two that matter most here: a zero exponent gives 1, and a negative exponent means a reciprocal. In Week 3 you learned function notation and the range. In Week 4 you met lines, which change by the same amount at every step, and in Week 10 you met horizontal asymptotes. This week the variable itself climbs up into the exponent. Next week, in Week 14, Logarithmic Functions, you'll learn to run this week's rule backwards.
I'll take eight ideas in order, from the difference between adding and multiplying all the way to the strange and useful number called e.
1. Compare adding with multiplying
Picture two coin jars. Into the first one, you drop 3 coins every day. The second one doubles whatever it holds every day. Soon they look nothing alike.
Take the line from Week 4, 3x plus 4. At x equals 0, 1, 2, and 3, it gives 4, 7, 10, and 13. Subtract each output from the next one and you get 3 every time. That's the line's constant difference, its slope.
Now take a new function, 3 times 2 to the x. At the same inputs, 0, 1, 2, and 3, it gives 3, 6, 12, and 24. Subtract neighbors and you get 3, then 6, then 12. No constant difference there. But divide neighbors: 6 divided by 3 is 2, 12 divided by 6 is 2, and 24 divided by 12 is 2. That's a constant ratio: each output divided by the output before it gives the same number every time the input goes up by 1.
So here's the test I want you to carry. Subtract neighbors and divide neighbors. Matching differences mean a line. Matching ratios mean an exponential. At x equals 0 the line is ahead, 4 to 3. By x equals 3 the exponential is far ahead, 24 to 13.
Pause and try this. The outputs 3, 6, 12, 24 go up fast. Is that table a line with a steep slope, or an exponential?
Count the differences first, then the ratios.
Here's the answer. It's an exponential, because the differences 3, 6, and 12 keep changing while every ratio is 2.
2. Read the initial value and the growth factor
Back to the doubling jar. Two numbers tell its whole story: how many coins it starts with, and what you multiply by each day. An exponential function packs exactly those two numbers into its formula.
An exponential function is a function of the form f of x equals a times b to the x, with a not zero, b positive, and b not equal to 1. The input sits in the exponent, so each one-unit step in the input multiplies the output by the same factor b. The number a is the initial value, the output when the input is 0, because b to the zero is 1. The base b is the growth factor, the number each output is multiplied by when the input goes up by 1.
For 3 times 2 to the x, the initial value is 3 and the growth factor is 2. The 2 is the base that wears the exponent, and the 3 out front is a coefficient.
Why can't b be 1? Because 1 to any power is still 1, so the outputs would never change, and you'd have a flat line.
Pause and try this. In 3 times 2 to the x, which number is the starting amount and which is the multiplier?
Find the number that wears the exponent.
Here's the answer. The starting amount is 3 and the multiplier is 2, because the exponent sits on the 2, and at x equals 0 the output is 3.
3. Work the exponent first: positive, zero, and negative inputs
Here's where Week 1's order of operations earns its keep. In 3 times 2 to the x, the exponent belongs to the 2 alone. So you raise 2 to the power first, and only then multiply by 3.
At x equals 2, the exponent goes first: 2 squared is 4. Then multiply: 3 times 4 is 12. At x equals 0, 2 to the zero is 1, and 3 times 1 is 3, the initial value again.
A negative exponent means a reciprocal. So 2 to the negative 2 is 1 over 2 squared, which is one fourth. Then 3 times one fourth is three fourths, or 0.75. At x equals negative 3, you get 3 times one eighth, which is three eighths, or 0.375. Small numbers, but positive ones.
Here's the trap to dodge. It's tempting to multiply 3 times 2 first and call it 6 to the x. Test it at x equals 2. Six squared is 36, but the real value is 12. Since 36 isn't 12, 3 times 2 to the x is not 6 to the x. Exponent first, then multiply.
Pause and try this. Someone says f of negative 2 for 3 times 2 to the x must be negative, because the input is negative. What is it really?
Think reciprocal, not negative.
Here's the answer. It's 0.75, because 2 to the negative 2 is one fourth, and 3 times one fourth is three fourths.
4. Use the base to tell growth from decay
Now flip the story. Picture a 100-gram sample of a substance that loses half its mass every period. After one period, 50 grams remain. After two, 25 grams. After three, 12.5 grams. After four, 6.25 grams. Each step multiplies by one half, so this is an exponential too. It shrinks instead.
Here are the words. Exponential growth is the behavior of a times b to the x, with a positive, when the base b is greater than 1: the outputs get larger by the same factor at every step. Exponential decay is the behavior when the base is between 0 and 1: the outputs shrink by the same factor at every step. A base between 0 and 1 is called a decay factor, the fraction that remains after each step. And a half-life is the length of time it takes a decaying amount to fall to half its size. The sample's decay factor is one half, so each period is one half-life.
Let's work one value. After three half-lives, the mass is 100 times one half cubed. One half cubed is one eighth, and 100 divided by 8 is 12.5 grams.
Now the trap. For 3 times 2 to the x, the output at negative 2 is 0.75, and some students see that small number and call the function decay. But growth or decay is decided by the base, not by one small output. Read that table left to right: 0.375, 0.75, 1.5, 3, and up from there. The outputs grow at every step, because the base, 2, is greater than 1. The base decides.
Pause and try this. Is 3 times 2 to the x growth or decay, given that its output at negative 2 is only 0.75?
Look at the base, not the size of the output.
Here's the answer. It's growth, because its base, 2, is greater than 1, so every step to the right doubles the output.
5. Read the graph from the formula
For an exponential, three graph features come straight from the formula.
First, the y-intercept. Set x equal to 0 and you get the initial value. So the graph of 3 times 2 to the x crosses the vertical axis at the point 0 comma 3, and the sample's graph crosses at 0 comma 100.
Second, the horizontal asymptote. Walk left through the table: 3, then 1.5, then 0.75, then 0.375. Each step halves the output. The outputs creep toward 0, but half of a positive number is still positive, so they never get there. The x-axis, the line y equals 0, is a horizontal asymptote. Picture the curve on the far left lying almost flat, a hair above the horizontal axis, never touching it.
Third, the shape. A growth graph rises everywhere, a decay graph falls everywhere, and neither one turns around.
Put those together and you get the domain and the range. You can raise 2 to any real power, so the domain is all real numbers. Every output is positive, so the range is every positive number, from 0 to infinity, with 0 itself left out.
Graph 3 times 2 to the x and the line 3x plus 4 together in Desmos or GeoGebra. The curve starts below the line at x equals 0, where 3 is less than 4. It passes the line before x equals 3, where 24 is more than 13.
Pause and try this. Can the graph of 3 times 2 to the x ever cross the horizontal axis?
Think about the sign of every output.
Here's the answer. No, because every output is a positive number times a power of 2, and every power of 2 is positive, so the curve stays above the axis and only approaches it.
6. Turn a percent rate into a factor
Real growth is usually given as a percent. A town grows 3% a year. A phone loses 25% of its value a year. To use the formula, you turn the percent into a factor.
Write the rate as a decimal: 3% is 0.03. For growth, the factor is 1 plus the rate, because you keep the whole amount and add the percent. For decay, it's 1 minus the rate, because you keep the whole amount and lose the percent.
So a town of 800 people growing 3% a year has a factor of 1.03, and its model is 800 times 1.03 to the t, with t in years. After ten years, using the pinned value of 1.03 to the tenth, about 1.3439, the town has about 1,075 people.
The phone goes the other way. A 1,200-dollar phone that loses 25% each year has a factor of 1 minus 0.25, which is 0.75. After one year it's worth 1,200 times 0.75, which is 900 dollars. After two years it's worth 900 times 0.75, which is 675 dollars.
The classic slip is writing the factor as 0.03, which would shrink the town to 3% of its size every year. You add the rate to 1.
Pause and try this. The phone loses 25% of its value a year. What factor do you multiply by each year, 0.25 or 0.75?
Ask what fraction of the value stays.
Here's the answer. It's 0.75, because the phone keeps 75% of its value each year, and 1 minus 0.25 is 0.75.
7. Compare compound and simple interest
Money in a savings account is the classic exponential. The principal is the amount of money first deposited or borrowed, before any interest is added. Simple interest is interest paid only on the original principal, the same dollar amount every year. Compound interest is interest added to the balance at the end of each period, so later interest is earned on earlier interest too. A compounding period is the time between those additions.
Take 1,000 dollars at 6% compounded once a year. The factor is 1.06. After one year, 1,000 times 1.06 is 1,060 dollars. After two years, 1,060 times 1.06 is 1,123.60 dollars.
Simple interest pays 6% of the original 1,000 dollars each year, which is 60 dollars. After two years that's 1,120 dollars. Compounding earned 3.60 dollars more. Here's why: the first year's 60 dollars of interest earned 6% in year two, and 6% of 60 is 3.60. Interest on interest.
More compounding periods earn a little more. Put 2,000 dollars at 4% compounded twice a year. Each half-year pays 2%. After the first half-year, you have 2,040 dollars. After the second, 2,080.80 dollars. Compounded once a year, it would be 2,080 dollars, so twice a year earns 80 cents more.
Pause and try this. After two years at 6%, compound interest gives 1,123.60 dollars and simple interest gives 1,120 dollars. Where does the extra 3.60 dollars come from?
Follow the first year's 60 dollars.
Here's the answer. It's the interest on the first year's interest, because 60 dollars earning 6% in year two makes 3.60 dollars.
8. Find the natural base e and compound continuously
What if a bank compounded every month, every day, every instant? The balance doesn't run off to infinity. It levels off, at one of the most important numbers in mathematics.
Look at 1 plus 1 over n, raised to the n. When n is 1, you get 2. When n is 2, you get 1.5 squared, which is 2.25. When n is 12, it's about 2.61. When n is 365, about 2.71. When n is 1,000,000, about 2.72. The values climb, but they level off. The number they level off at is the natural base e, about 2.71828. Like pi, e is irrational, so its decimal never ends and never repeats. You use the e key on your calculator, not a rounded 2.7.
Interest compounded at every instant is called continuous compounding, and the balance is the principal times e raised to the rate times the time. Take 1,500 dollars at 5% for 4 years. The rate times the time is 0.05 times 4, which is 0.2. The pinned value of e to the 0.2 is about 1.2214. Multiply by 1,500 and the balance is about 1,832.10 dollars.
One more pattern to read. Any continuous model, a starting amount times e to the k times t, grows when k is positive and decays when k is negative. In the interest model, k is the rate, 0.05, so it grows. In a model with negative 0.15 times t up in the exponent, k is negative 0.15, so it decays. This week you only read that sign.
Pause and try this. Is e a variable you solve for, or a fixed number like pi?
Think about the table that levels off.
Here's the answer. It's a fixed number like pi, because the values of 1 plus 1 over n, raised to the n, settle at the same constant, about 2.71828, every time.
Closing
Let's pull it together, in the order we took it. A line adds the same amount at every step, and an exponential multiplies by the same factor. In a times b to the x, a is the starting output and b is the multiplier. You do the exponent first, so 3 times 2 to the x is never 6 to the x, and a negative input gives a small positive output, never a negative one. A base above 1 means growth, and a base between 0 and 1 means decay; one small output doesn't change that. The graph crosses the vertical axis at the initial value and hugs the horizontal axis without touching it. A percent rate becomes a factor of 1 plus the rate for growth or 1 minus the rate for decay. Compound interest multiplies each period and earns interest on interest, while simple interest adds the same amount. And e, about 2.71828, is the fixed number behind continuous compounding.
Next in this module, you'll find the slide deck, then the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Week 14, Logarithmic Functions, you'll learn the rule that runs an exponential backwards and asks which exponent gives a certain output.
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