Linear and Exponential Models from Tables, SAT math lesson

One savings jar gets 10 dollars every week. A colony of bacteria doubles every hour. After a few steps the two tables look nothing alike, and the reason is simple. The jar adds the same amount each step, while the colony multiplies by the same factor each step. A table of values often hides which of these is happening until you compare the outputs in the right way, and that comparison is what many SAT questions about models are really testing.

See also Exponential Functions and Growth/Decay in Precalculus and Choosing and Interpreting Function Models in Precalculus.

Key terms

Equal input intervals

Definition The inputs $x_0 < x_1 < x_2 < \dots$ of a table are equally spaced when consecutive inputs differ by the same positive number $h$ , so that

x_{k+1} - x_k = h \quad \text{for every } k.

The number $h$ is the step. Comparisons of consecutive outputs, by subtraction or by division, only reveal a pattern when they are made over input intervals of equal length.

In plain words Before you compare outputs, check that the inputs move in equal steps. If they do not, the raw differences can mislead you.

Example Look at this table.

$x$	$0$	$1$	$3$	$4$
$y$	$5$	$8$	$14$	$17$

The output differences are $3$ , $6$ and $3$ , which seem uneven. But the middle step covers $2$ units of $x$ , not $1$ . Dividing each difference by its step gives $3$ per unit every time, so the points do lie on one line, $y = 5 + 3x$ . Dividing by the step only fixes differences. Ratios over unequal steps would need roots, so compare outputs over equal steps whenever you can.

Constant difference

Definition A table with equally spaced inputs has a constant difference when consecutive outputs differ by the same number $d$ ,

y_{k+1} - y_k = d \quad \text{for every } k.

The rate of change per unit of input is then $d/h$ , where $h$ is the step.

In plain words Each step adds the same amount. The amount can be negative, which means the quantity drops by the same amount each step.

Example A tank is measured every $2$ minutes.

$t$ (minutes)	$0$	$2$	$4$	$6$
$V$ (liters)	$40$	$46$	$52$	$58$

Every difference is $6$ liters, and each covers $2$ minutes. So the volume rises by $6/2 = 3$ liters per minute, not $6$ liters per minute. Mixing up the step and the unit is the most common slip with tables like this.

Linear model

Definition A linear model has the form

f(x) = b + mx,

where $b$ is the value at $x = 0$ and $m$ is the constant rate of change. For every $x$ and every step $h$ ,

f(x + h) - f(x) = mh,

so over equal input steps a linear function always has a constant difference. In the other direction, if a table has equally spaced inputs and a constant difference, its points all lie on one line, so a linear model fits the table exactly.

In plain words Linear means a fixed amount is added for each unit of input. The graph is a straight line that starts at $b$ when $x = 0$ .

Example For the tank above, the starting value is $40$ and the rate is $3$ liters per minute, so

V(t) = 40 + 3t.

Check the last row. At $t = 6$ the model gives $40 + 18 = 58$ , which matches the table.

Constant ratio

Definition A table with equally spaced inputs and outputs that are all nonzero has a constant ratio when each output divided by the one before it gives the same number $r$ ,

\frac{y_{k+1}}{y_k} = r \quad \text{for every } k.

The ratio is only defined when the denominator $y_k$ is not zero, so a table containing an output of $0$ cannot have a constant ratio in this sense.

In plain words Each step multiplies by the same factor. For positive outputs, a factor above $1$ means growth and a factor between $0$ and $1$ means decay. A negative ratio makes the outputs flip sign at every step, which no exponential model $a \cdot b^x$ with $b > 0$ can do.

Example A town’s number of bike rentals is recorded each year.

$t$ (years)	$0$	$1$	$2$	$3$
$R$	$200$	$240$	$288$	$345.6$

The differences are $40$ , $48$ and $57.6$ , so they are not constant. The ratios are

\frac{240}{200} = \frac{288}{240} = \frac{345.6}{288} = 1.2,

so each year the number is multiplied by $1.2$ , which is growth of $20$ percent per year.

Exponential model

Definition An exponential model has the form

f(x) = a \cdot b^x,

where $a \ne 0$ is the value at $x = 0$ and the growth factor $b$ satisfies $b > 0$ and $b \ne 1$ . For every $x$ and every step $h$ ,

\frac{f(x + h)}{f(x)} = b^h,

so over equal input steps an exponential function always has a constant ratio. The model shows growth when $b > 1$ and decay when $0 < b < 1$ .

In plain words Exponential means a fixed factor is applied for each unit of input. The graph is a curve that keeps its starting value $a$ at $x = 0$ and, when $a > 0$ and $b > 1$ , bends upward faster and faster.

Example For the rentals, the starting value is $200$ and the yearly factor is $1.2$ , so

R(t) = 200(1.2)^t.

At $t = 3$ this gives $200 \cdot 1.728 = 345.6$ , which matches the table.

Per-unit factor from a longer step

Definition Suppose the inputs of a table step by $h$ units and the outputs have constant ratio $r > 0$ . An exponential model $f(x) = a \cdot b^x$ that fits the table has $b^h = r$ , so the per-unit factor is

b = r^{1/h},

and the model can be written as

f(x) = a \cdot r^{x/h}.

In plain words If the table jumps several units at a time, the ratio you see is the factor for the whole jump. To get the factor for one unit, take the $h$ th root.

Example An account is checked every $3$ years.

$t$ (years)	$0$	$3$	$6$
$A$ (dollars)	$500$	$540$	$583.20$

The ratio is $1.08$ for each $3$ year step, so the model is $A = 500(1.08)^{t/3}$ . The yearly factor is

b = 1.08^{1/3} \approx 1.026,

about $2.6$ percent per year, not $8$ percent. Writing $500(1.08)^t$ would claim $8$ percent every single year, which is far too fast.

Choosing a model from a table

Definition To choose between a linear and an exponential model, first confirm equal input steps, then compute consecutive differences and, when no output is zero, consecutive ratios. A constant difference supports a linear model and a constant positive ratio other than $1$ supports an exponential model. A finite table only shows that a model fits the listed points. It does not prove what happens at inputs the table does not list, and different functions can agree on every row of a table.

In plain words The table is evidence, not proof. Pick the model whose pattern matches, and say that you are assuming the pattern continues.

Example The table with $x = 0, 1, 2$ and $y = 1, 2, 4$ has ratios $2$ and $2$ , so $y = 2^x$ fits it. The quadratic

y = \frac{x^2 + x + 2}{2}

also gives $1$ , $2$ and $4$ at those inputs. At $x = 3$ the first model gives $8$ and the second gives $7$ . Only the context, such as a colony that doubles every hour, tells you which one to trust beyond the table.

Check yourself

The table shows values of two functions. Which statement is true? (A) The function $f$ is linear and $g$ is exponential. (B) The function $f$ is exponential and $g$ is linear. (C) Both functions are exponential. (D) Both functions are linear.

$x$ $0$ $1$ $2$ $3$
$f(x)$ $3$ $6$ $12$ $24$
$g(x)$ $3$ $6$ $9$ $12$

$x$	$0$	$1$	$2$	$3$
$f(x)$	$3$	$6$	$12$	$24$
$g(x)$	$3$	$6$	$9$	$12$

Show the answer

The correct choice is (B). The values of $f$ have ratio $2$ at every step, so $f(x) = 3 \cdot 2^x$ . The values of $g$ have difference $3$ at every step, so $g(x) = 3 + 3x$ . Both start at $3$ and both go from $3$ to $6$ in the first step, which is why a quick glance at one step cannot tell them apart and choices (C) and (D) are tempting. Choice (A) swaps the two patterns, reading the doubling of $f$ as a fixed amount added.

A biologist counts bacteria every $4$ hours and records $800$ at $t = 0$ , $1200$ at $t = 4$ and $1800$ at $t = 8$ , where $t$ is in hours. Which equation models the count $P$ ? (A) $P = 800(1.5)^t$ (B) $P = 800(1.5)^{t/4}$ (C) $P = 800(1.5)^{4t}$ (D) $P = 800 + 100t$

Show the answer

The correct choice is (B). The ratios are $1200/800 = 1.5$ and $1800/1200 = 1.5$ , and each covers $4$ hours, so the factor $1.5$ applies once every $4$ hours. Choice (A) would multiply by $1.5$ every hour, and (C) every quarter hour. Choice (D) matches the first two counts but gives $1600$ at $t = 8$ , not $1800$ .

The value of an investment is $1000$ dollars at $t = 0$ , $1210$ dollars at $t = 2$ and $1464.10$ dollars at $t = 4$ , where $t$ is in years. The value is modeled by $V = 1000b^t$ . What is the value of $b$ ?

Show the answer

The answer is $1.1$ . The ratio over each $2$ year step is $1210/1000 = 1.21$ , so $b^2 = 1.21$ and $b = \sqrt{1.21} = 1.1$ , using the positive root because a growth factor is positive. A check with the last row gives $1000(1.1)^4 = 1464.1$ . Entering $1.21$ would be the factor for two years, not one.

Linear and Exponential Models from Tables

Theory, part 73

Practice, part 74

Key terms

Equal input intervals

Constant difference

Linear model

Constant ratio

Exponential model

Per-unit factor from a longer step

Choosing a model from a table

Check yourself

How this lesson connects

Builds on

Used later in