Exponential functions model situations where the outputs over equal-length input intervals are proportional. When the inputs are whole numbers, the situation is repeated multiplication of a constant factor applied to an initial value:
Example. A colony starts with 200 cells and grows by 50% each day. Each day multiplies the count by 1.5, so the counts are 200, 300, 450, 675, … The ratios 300/200, 450/300, and 675/450 are all 1.5. The model is P(t) = 200(1.5)t.
Pick a situation and the number of whole-number steps n. Each step multiplies the output by the same factor, so the outputs are proportional over equal-length intervals.
| day | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| amount | 200 | 300 | 450 | 675 | 1012.5 |
| ratio | — | 1.5 | 1.5 | 1.5 | 1.5 |
Sometimes the outputs in a data set are not proportional, but a constant added to (or subtracted from) every output reveals a proportional pattern. The constant is the vertical shift k in f(x) = abx + k.
Example. A cup of hot chocolate sits in a 22°C room. Its temperatures at hours 0, 1, 2, 3, 4 are 90, 83.2, 77.08, 71.572, 66.6148. These are not proportional. Subtract the room temperature, 22: 68, 61.2, 55.08, 49.572, 44.6148. Now each is 0.9 times the one before, so T(t) = 68(0.9)t + 22.
Each list is the output of a function at x = 0, 1, 2, 3, 4. First check whether the outputs are proportional. If not, slide k to subtract a constant from every output (to add 5, set k = −5), and look for a value of k that makes the new outputs proportional.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 90 | 83.2 | 77.08 | 71.572 | 66.615 |
| f(x) − 0 | 90 | 83.2 | 77.08 | 71.572 | 66.615 |
| ratios | — | 0.924 | 0.926 | 0.928 | 0.931 |
An exponential model can be built from an initial value and a ratio, or from two input-output pairs. With two pairs (x₁, y₁) and (x₂, y₂), the initial value a and the base b come from a system of equations:
Divide the second equation by the first. The a cancels, leaving bx₂ − x₁ = y₂ / y₁. Take the root to find b, then substitute back to find a.
Example. For (1, 6) and (3, 24): b2 = 24/6 = 4, so b = 2 (the base is positive). Then a·2 = 6, so a = 3. The model is f(x) = 3·2x.
Two input-output pairs give two equations, a·bx₁ = y₁ and a·bx₂ = y₂. Divide the second by the first to get b, then back-substitute to get a.
A model can be built by applying transformations to f(x) = abx, chosen to match the characteristics of the situation or data. The most useful form is:
| Change | Effect on the graph of bx |
|---|---|
| a > 1 or 0 < a < 1 | vertical stretch or shrink |
| a < 0 | reflection over the x-axis (concave down) |
| x − h | horizontal shift by h (equivalent to a vertical dilation by b−h) |
| + k | vertical shift up by k; the graph levels off toward y = k |
Build f(x) = a·bx − h + k from the parent function bx. The dashed line is y = k. Notice that the horizontal shift h can be absorbed into the multiplier: a·bx − h = (a·b−h)·bx.
Real data is rarely perfect. For a data set, an exponential function model can be constructed with technology using exponential regression. A calculator or graphing tool finds the function y = a·bx that best fits the data.
A graphing calculator finds the exponential function y = a·bx that best fits a data set. The tool does what the calculator does. The blue curve is the regression model and the gold dots are the data.
| hours | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| data | 2.1 | 3.1 | 5.3 | 8 | 13.4 | 20.5 | 34.2 |
| model | 2.03 | 3.24 | 5.17 | 8.25 | 13.15 | 20.98 | 33.48 |
The number e ≈ 2.718 is often used as the base in exponential functions that model real situations. A model written with the natural base looks like:
This is still an exponential function. By the power property from Lesson 2.4, a·ekx = a·(ek)x, so the ordinary base is b = ek. For example, 100·e0.03t = 100(1.0305)t.
The number e ≈ 2.718 is often used as the base of models: f(x) = a·ekx. Because of the power property, a·ekx = a·(ek)x, an ordinary exponential function with base b = ek.
For a model f(x) = a·bx, the base b is the growth factor for each unit change in the input. It is related to a percent change in context:
| Situation | Factor b |
|---|---|
| 8% increase per year | 1.08 |
| 12% decrease per year | 0.88 |
| doubles each step (100% increase) | 2 |
| halves each step (50% decrease) | 0.5 |
In f(x) = a·bx, the base b is the growth factor for each unit step. A percent change r gives b = 1 + r/100. Slide the percent change (the initial value here is 100).
Equivalent forms of an exponential function can reveal different properties. If d is the number of days, the base of f(d) = 2d says the quantity increases by a factor of 2 every day. The equivalent form f(d) = (27)d/7 = 128d/7 says it increases by a factor of 128 every week.
A quantity is modeled by f(d) = 2d, where d is the number of days. It doubles every day. Pick a time unit to see the equivalent form (2u)d/u and the factor for that unit.
| d (days) | 0 | 7 | 14 |
|---|---|---|---|
| 2d | 1 | 128 | 16,384 |
| (128)d/7 | 1 | 128 | 16,384 |
Exponential models can be used to predict values of the dependent variable, depending on the contextual constraints on the domain. They can also be used to determine values of the independent variable: for what input does the model reach a given output?
A model gives the output for any allowed input. A graph or table can also answer the reverse question: for what input does the model reach a target? Use the intersection of the curve with the line y = target.