In Lesson 2.1 an arithmetic sequence an = a0 + dn gave you a set of separate points, one for each whole number n. Now ask a new question: what if the input could be any real number? The answer is a linear function f(x) = b + mx. The two have the same structure:
| Arithmetic sequence | Linear function | |
|---|---|---|
| Form | an = a0 + dn | f(x) = b + mx |
| Initial value | a0 | b (the output at x = 0) |
| Constant rate of change | d (common difference) | m (slope) |
| Built from | an initial value plus repeated addition of the constant rate of change | |
Example. The sequence an = 3 + 2n has terms 3, 5, 7, 9, … The function f(x) = 3 + 2x gives the same outputs at x = 0, 1, 2, 3, but it also has values in between: f(0.5) = 4 and f(2.5) = 8.
The blue dots are the arithmetic sequence an = b + mn, which exists only at whole numbers. The line is the linear function f(x) = b + mx, which exists at every real number. Slide x to a whole number and then to a number in between.
In 2.1 you wrote an = ak + d(n − k) when you knew a difference d and one term ak. A linear function has the same form. If you know the slope m and a point (xi, yi) on the line:
Example. A line has slope 2 and passes through (3, 9). Then f(x) = 9 + 2(x − 3). Distribute: 9 + 2x − 6 = 2x + 3, so f(x) = 3 + 2x and the initial value is b = 3. This is the function from the last section. Starting from any known point gives the same line.
Choose a slope m and a known point (xi, yi). The point form and the slope-intercept form describe the same line.
Do the same thing for a geometric sequence gn = g0rn. Let the input be any real number and you get an exponential function f(x) = a·bx. Again the structure matches:
| Geometric sequence | Exponential function | |
|---|---|---|
| Form | gn = g0rn | f(x) = abx |
| Initial value | g0 | a (the output at x = 0) |
| Constant ratio | r (common ratio) | b (the base) |
| Built from | an initial value times repeated multiplication by the constant ratio | |
Example. The sequence gn = 3·2n has terms 3, 6, 12, 24, … The function f(x) = 3·2x gives the same outputs at x = 0, 1, 2, 3, and also has values in between. For example f(0.5) = 3·20.5 ≈ 4.243 and f(2.5) = 3·22.5 ≈ 16.971.
The orange dots are the geometric sequence gn = a·bn, which exists only at whole numbers. The curve is the exponential function f(x) = a·bx, which exists at every real number. Slide x to a whole number and then to a number in between.
In 2.1 you wrote gn = gkrn − k from a ratio r and one term gk. An exponential function has the same form. If you know the ratio b and a point (xi, yi):
Example. An exponential function has ratio 2 and passes through (2, 12). Then f(x) = 12·2x − 2. Since 2x − 2 = 2x / 4, this is f(x) = 3·2x, so the initial value is a = 3. That is the function from the last section.
Choose a ratio b and a known point (xi, yi). The point form and the a·bx form describe the same curve.
The formulas look almost the same, but the domain (the set of allowed inputs) is not. A sequence takes only whole numbers. A linear or exponential function takes every real number. That is why a sequence graphs as separate dots while its function graphs as one connected line or curve.
| Model | Domain | Graph |
|---|---|---|
| an = 3 + 2n or gn = 3·2n | whole numbers 0, 1, 2, … | separate points |
| f(x) = 3 + 2x or f(x) = 3·2x | all real numbers | connected line or curve |
Pick a model, then slide the input, including negative numbers and numbers in between. A sequence only accepts the whole numbers 0, 1, 2, …. A linear or exponential function accepts every real number.
How can you tell from a table which kind of function you have? Look at the outputs over equal-length input intervals:
| Over equal-length input intervals, the outputs… | The function is… |
|---|---|
| change at a constant rate (the same amount added) | linear |
| change proportionally (the same ratio) | exponential |
Example. For f(x) = 2x + 1 with step 0.5, the outputs are 1, 2, 3, 4, 5. Every interval adds 1, so it is linear. For g(x) = 3·2x with step 0.5, every interval multiplies by 20.5 ≈ 1.414, so it is exponential. The step does not have to be 1. It only has to be the same for every interval.
Pick a function and a spacing for the inputs. The tool finds the change (output − previous output) and the ratio (output ÷ previous output) over each interval. Then it tells you what the pattern means. Try the uneven spacing too.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| output | 1 | 3 | 5 | 7 | 9 |
| change | — | + 2 | + 2 | + 2 | + 2 |
| ratio | — | ×3 | ×1.667 | ×1.4 | ×1.286 |
Linear and exponential functions can both be described by an initial value and a constant that controls the change. The difference is the operation: linear functions are based on addition, and exponential functions are based on multiplication.
| Linear | Exponential | |
|---|---|---|
| Form | f(x) = b + mx | f(x) = abx |
| Initial value | b | a |
| Constant | m (slope) | b (ratio) |
| When x goes up by 1 | add m to the output | multiply the output by b |
| Equal-length intervals | constant change | proportional change |
Both functions start at the same initial value. The blue line adds m for every 1 unit of x. The orange curve multiplies by b for every 1 unit of x.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| linear | 4 | 7 | 10 | 13 | 16 | 19 |
| change | — | + 3 | + 3 | + 3 | + 3 | + 3 |
| exponential | 4 | 6 | 9 | 13.5 | 20.25 | 30.375 |
| ratio | — | ×1.5 | ×1.5 | ×1.5 | ×1.5 | ×1.5 |
Arithmetic sequences, linear functions, geometric sequences, and exponential functions all share one more property: each can be pinned down by two distinct sequence or function values. The two values tell you the constant, and then the point form finishes the job.
Example. The values are f(1) = 6 and f(3) = 24.
Linear: m = (24 − 6) / (3 − 1) = 9, so f(x) = 6 + 9(x − 1) = 9x − 3.
Exponential: b2 = 24 / 6 = 4, so b = 2 (the base is positive). Then f(x) = 6·2x − 1 = 3·2x.
Choose two input-output values, (x1, y1) and (x2, y2). Exactly one line and exactly one exponential curve pass through both. (Outputs are kept positive so an exponential curve can exist.)