Unit 2 · Topic 2.2

Change in Linear and Exponential Functions

Over equal-length input intervals, a linear function adds a constant difference and an exponential function multiplies by a constant ratio.

Practice 2: Multiple RepresentationsPractice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

A music app has 500 users. One plan adds 200 users a month; another doubles the users every month. After a year the first plan has 2,900 users and the second has over 2 million. The difference is adding versus multiplying.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is the common ratio of the geometric sequence $6, 18, 54, \ldots$ ?

$3$
$12$
$6$
$\frac{1}{3}$

Show the answer

$\frac{18}{6} = 3$ .

Correct: $3$ :
$12$ :
$6$ :
$\frac{1}{3}$ :

2. What is the average rate of change of $f$ on $[1, 4]$ if $f(1) = 2$ and $f(4) = 14$ ?

$4$
$12$
$3$
$\frac{1}{4}$

Show the answer

$\frac{14 - 2}{4 - 1} = 4$ .

Correct: $4$ :
$12$ :
$3$ :
$\frac{1}{4}$ :

Part 4 · See it

See it first

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. — Over equal steps, $f(x) = 4 + 6x$ adds $6$ each time while $g(x) = 4(1.5)^x$ multiplies by $1.5$ . Both start at $4$ ; the exponential function falls behind at first, then passes the line between $x = 5$ and $x = 6$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Check that the input intervals have equal lengthonly then compare the outputs directly
Equal differences over the equal intervalsa linear function, $f(x) = b + mx$
Equal ratios (not $0$ or $1$ ) over the equal intervalsan exponential function, $f(x) = ab^x$ , or $f(x) = y_1 r^{x - x_1}$ through a known point

Part 6 · Key ideas

Key ideas

Linear functions extend arithmetic sequences to all real inputs; exponential functions extend geometric sequences.
Linear: over equal-length input intervals, the outputs change by equal amounts. Exponential: over equal-length input intervals, the outputs change by equal factors (they are proportional).
An increasing exponential function eventually passes any increasing linear function.

Part 7 · Misconception

A common mistake

The wrong idea: The outputs increase faster and faster, so the function is exponential.

What actually happens: Say both halves with numbers: the input intervals have equal length, and the outputs over them have a constant ratio. Faster and faster growth also fits a quadratic.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Bacteria in a culture

Number of bacteria B(t), in thousands, t hours after a culture is started
$t$ (hours)	0	2	4	6	8
$B(t)$ (thousands)	5	15	45	135	405

1. Which statement correctly justifies the type of function that fits the table?

Exponential: the outputs increase faster and faster as $t$ increases, so the growth speeds up
Exponential: over equal input intervals of $2$ hours, each output is $3$ times the one before
Linear: over equal input intervals of $2$ hours, the outputs increase by a larger amount each time
Exponential: each output in the table is $3$ times the output in the column before it

Show the answer

Both halves are needed: the input intervals have equal length ( $2$ hours each), and the outputs over them have a constant ratio ( $\frac{15}{5}$ $= \frac{45}{15}$ $= \frac{135}{45}$ $= \frac{405}{135}$ $= 3$ ). That is the pattern of an exponential function.

Exponential: the outputs increase faster and faster as $t$ increases, so the growth speeds up: Faster and faster growth fits many functions (a quadratic too); it is not a justification.
Correct: Exponential: over equal input intervals of $2$ hours, each output is $3$ times the one before: Equal-length input intervals and a constant ratio: both halves, with the numbers.
Linear: over equal input intervals of $2$ hours, the outputs increase by a larger amount each time: A linear function needs equal differences, and these are $10, 30, 90, 270$ .
Exponential: each output in the table is $3$ times the output in the column before it: This leaves out that the input intervals are equal in length, half of the justification.

2. Write an expression for $B(t)$ , in thousands, that fits the table.

Type an expression for $B(t) =$ .

Show the answer

The outputs triple every $2$ hours starting from $B(0) = 5$ , so after $t$ hours there have been $\frac{t}{2}$ triplings: $B(t) = 5(3)^{t/2}$ . Check: $B(4) = 5(3)^2 = 45$ .

Answer: $B(t) =$ $5(3)^{\frac{t}{2}}$

3. If the pattern in the table continues, what is $B(10)$ , in thousands?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Two more hours is one more tripling: $B(10) = 3 \cdot 405 = 1215$ thousand.

Answer: 1215

Table

Two functions at unequally spaced inputs

Values of p(x) and q(x) at x = 0, 1, 3 and 6; the input steps are 1, 2 and 3
$x$	0	1	3	6
$p(x)$	10	13	19	28
$q(x)$	4	8	32	256

4. The differences in $p(x)$ are $3$ , $6$ and $9$ . Which conclusion is correct?

$p$ is not linear, because the differences $3$ , $6$ , $9$ are not constant
$p$ is exponential, because its differences increase from step to step
$p$ can be linear: each difference divided by its input step is $3$
$p$ is linear, because its outputs increase as $x$ increases

Show the answer

The input steps are $1$ , $2$ and $3$ , not equal, so raw differences cannot be compared. Divide each by its step: $\frac{3}{1} = \frac{6}{2} = \frac{9}{3} = 3$ . A constant rate of change of $3$ fits $p(x) = 10 + 3x$ .

$p$ is not linear, because the differences $3$ , $6$ , $9$ are not constant: Constant differences are the test only over equal-length input intervals.
$p$ is exponential, because its differences increase from step to step: The differences grow because the steps grow; the rate is constant.
Correct: $p$ can be linear: each difference divided by its input step is $3$ : Unequal steps: compare rates, not raw differences.
$p$ is linear, because its outputs increase as $x$ increases: Increasing outputs do not decide the type.

5. Which statement about $q$ is supported by the table?

$q$ is not exponential, because the ratios $2$ , $4$ , $8$ are not equal
$q$ is linear, because each ratio of outputs is a whole number
$q$ is exponential with a ratio of $8$ for each unit increase in $x$
$q$ can be exponential: outputs double per $1$ -unit increase

Show the answer

Over steps of $1$ , $2$ and $3$ units the ratios are $2 = 2^1$ , $4 = 2^2$ and $8 = 2^3$ . That is a doubling for every $1$ unit, so $q(x) = 4(2)^x$ fits.

$q$ is not exponential, because the ratios $2$ , $4$ , $8$ are not equal: The ratios differ only because the steps differ.
$q$ is linear, because each ratio of outputs is a whole number: Whole-number ratios do not make a function linear.
$q$ is exponential with a ratio of $8$ for each unit increase in $x$ : $8$ is the ratio over the $3$ -unit step.
Correct: $q$ can be exponential: outputs double per $1$ -unit increase: Ratio $2^k$ over a step of $k$ units: doubling per unit.

6. An exponential function $f$ has $f(2) = 18$ , and its outputs triple for each increase of $1$ in $x$ . Write $f(x)$ using the point $(2, 18)$ .

Type an expression for $f(x) =$ .

Show the answer

From a known point $(x_1, y_1)$ with ratio $r$ per unit: $f(x)$ $= y_1 r^{x - x_1}$ $= 18(3)^{x - 2}$ . Check: $f(3) = 54$ .

Answer: $f(x) =$ $18(3)^{x - 2}$

Part 9 · Summary

Summary

Over equal-length input intervals, a linear function adds a constant difference and an exponential function multiplies by a constant ratio. Justify a model type with both halves and the numbers; with unequal steps, compare rates or per-unit ratios instead.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic