Linear vs Exponential Change — AP® Precalculus Notes

Arithmetic and geometric sequences are defined only at whole-number positions. Fill in every real input between them and you get two of the most useful function types in the course: linear functions, which extend arithmetic sequences, and exponential functions, which extend geometric ones. This page is about telling them apart from a table, and about justifying your answer the way the exam scores it.

Adding versus multiplying

A linear function $f(x) = b + mx$ adds the same amount $m$ for each unit of input. An exponential function $f(x) = ab^x$ (with $a \ne 0$ , $b > 0$ and $b \ne 1$ ) multiplies by the same factor $b$ for each unit of input. If $b > 1$ the function shows exponential growth; if $0 < b < 1$ , exponential decay.

The line f(x) = 4 + 6x adding 6 per step and the exponential g(x) = 4(1.5)^x multiplying by 1.5 per step, from x = 0 to 6; the exponential passes the line between x = 5 and x = 6. — **Figure 1.** $f(x) = 4 + 6x$ adds $6$ per step; $g(x) = 4(1.5)^x$ multiplies by $1.5$ per step. LevlPrep original diagram.

In the figure, the line is ahead at first, but the exponential function's steps keep getting bigger ( $2$ , $3$ , $4.5$ , $6.75$ , ...) and it passes the line between $x = 5$ and $x = 6$ . An increasing exponential function eventually passes any increasing linear function.

The test, with both halves

Over equal-length input intervals:

What the outputs do	Function type	Example over steps of $2$
Change by a constant difference	Linear	$7, 13, 19, 25$ : add $6$ each time
Change by a constant ratio (not $0$ or $1$ ); the outputs are proportional	Exponential	$5, 15, 45, 135$ : multiply by $3$ each time

A full justification names both halves and uses the numbers: "The input intervals all have length $2$ , and the outputs over them have a constant ratio, $\frac{15}{5}$ $= \frac{45}{15}$ $= \frac{135}{45}$ $= 3$ , so an exponential function fits." Readers take the point away if either half is missing. "The outputs grow faster and faster" earns nothing, because a quadratic does that too.

Worked example 1. A culture has $5$ , $15$ , $45$ , $135$ and $405$ thousand bacteria at $t = 0, 2, 4, 6, 8$ hours. Choose a model and write it.

Equal intervals: each step is $2$ hours. Differences: $10, 30, 90, 270$ , not constant, so not linear. Ratios: $3, 3, 3, 3$ , constant, so exponential.

The output triples every $2$ hours, so after $t$ hours it has tripled $\frac{t}{2}$ times: $B(t) = 5(3)^{t/2}$ .

At $t = 1$ , half of a tripling period: $B(1) = 5(3)^{1/2} = 5\sqrt{3}$ . The output is multiplied by $\sqrt{3}$ each hour, because $\sqrt{3} \cdot \sqrt{3} = 3$ . Halfway in time is not halfway in value.

The trap: unequal input steps

If the inputs are $0, 1, 3, 6$ , the steps are $1$ , $2$ and $3$ , and raw differences or ratios mean nothing until you account for the step lengths.

Worked example 2. $p(x)$ is $10, 13, 19, 28$ and $q(x)$ is $4, 8, 32, 256$ at $x = 0, 1, 3, 6$ . What could each be?

For $p$ , the differences are $3, 6, 9$ . Divide each by its step: $\frac{3}{1} = \frac{6}{2} = \frac{9}{3} = 3$ . A constant rate of change: $p(x) = 10 + 3x$ fits.

For $q$ , the ratios are $2, 4, 8$ , which look unequal. But $2 = 2^1$ over $1$ unit, $4 = 2^2$ over $2$ units, $8 = 2^3$ over $3$ units: the output doubles for every unit. $q(x) = 4(2)^x$ fits.

An exponential function through a known point

If you know one point $(x_1, y_1)$ and the ratio $r$ per unit of input, count the steps from that point, just as for a geometric sequence: $f(x) = y_1 r^{x - x_1}$ . With $f(2) = 18$ and a ratio of $3$ : $f(x) = 18(3)^{x - 2}$ , so $f(3) = 54$ and $f(0) = 2$ .

Rates of change

For a linear function the average rate of change is the same on every interval. For an exponential function, the average rates of change over equal-length intervals have the same ratio as the outputs. In the bacteria table, the average rates on $[0, 2]$ , $[2, 4]$ and $[4, 6]$ are $5$ , $15$ and $45$ thousand per hour: each is $3$ times the one before, so the rate of change of $B$ is increasing and the graph is concave up.