Unit 1 · Topic 1.1

Change in Tandem

Two quantities that change together are described by direction and by how fast.

Practice 2: Multiple RepresentationsPractice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

A pond is deepest after the spring rains and shallowest in a dry month. Its depth and the date change together: when one moves, the other moves too. This first topic is about describing that kind of joint change in words a grader accepts: increasing or decreasing, and how fast.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. For $f(x) = (x^3 - 12x)/4$ , what is $f(2)$ ?

$-4$
$4$
$-2$
$0$

Show the answer

$(8 - 24)/4 = -16/4 = -4$ .

Correct: $-4$ :
$4$ :
$-2$ :
$0$ :

2. Which interval is $-2 < x \le 6$ ?

$(-2, 6]$
$[-2, 6)$
$(-2, 6)$
$[-2, 6]$

Show the answer

Parenthesis at the excluded end $-2$ , bracket at the included end $6$ .

Correct: $(-2, 6]$ :
$[-2, 6)$ :
$(-2, 6)$ :
$[-2, 6]$ :

Part 4 · See it

See it first

The graph of f(x) = (x³ − 12x)/4 on [−4, 4]. It increases from (−4, −4) to (−2, 4), decreases from (−2, 4) to (2, −4), then increases to (4, 4). Left of x = 0 the graph bends downward (concave down); right of x = 0 it bends upward (concave up). — $f(x) = (x^3 - 12x)/4$ increases on $[-4, -2]$ , decreases on $[-2, 2]$ and increases on $[2, 4]$ . It is concave down left of $x = 0$ and concave up right of it. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

For inputs $a < b$ in an interval, $f(a) < f(b)$ every time $f$ is increasing on that interval (and decreasing if $f(a) > f(b)$ every time)
The output changes by more over each equal step of inputthe rate of change is increasing and the graph is concave up
The output changes by less over each equal step (or by more negative amounts)the rate of change is decreasing and the graph is concave down
A function switches from decreasing to increasingits outputs reach a lowest value there, as the pond does at $t = 2$

Part 6 · Key ideas

Key ideas

Increasing on an interval: larger inputs give larger outputs there. Decreasing: larger inputs give smaller outputs.
The rate of change compares the change in output with the change in input: how fast the output moves.
Concave up: the rate of change is increasing (the graph bends upward). Concave down: the rate of change is decreasing.
Direction and concavity are separate: a function can be decreasing and concave up, like cooling coffee.

Part 7 · Misconception

A common mistake

The wrong idea: If a function is decreasing, its rate of change must be decreasing too.

What actually happens: The coffee's temperature falls, but its drops shrink: $-22, -17, -13, -9$ . The temperature is decreasing while its rate of change is increasing, so the graph is concave up.

Part 8 · Check yourself

Check yourself

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

Graph

Depth of a pond

1. The graph shows the depth $h(t)$ of a pond, in feet, $t$ months after the start of a year, for $0 \le t \le 8$ . On which interval is $h$ increasing?

$[0, 2]$
$[2, 6]$
$[4, 8]$
$[2, 10]$

Show the answer

$h$ is increasing on an interval when a larger input always gives a larger output there. The depth climbs from $h(2) = 2$ to $h(6) = 10$ and falls on both sides.

$[0, 2]$ : On $[0, 2]$ the graph falls from $10$ to $2$ : later times have smaller depths, so $h$ is decreasing there.
Correct: $[2, 6]$ : Right: from $t = 2$ to $t = 6$ , a later time always gives a greater depth.
$[4, 8]$ : $h$ rises only until $t = 6$ ; on $[6, 8]$ it falls from $10$ to $2$ .
$[2, 10]$ : $10$ is a depth (an output). The interval is in $t$ , the input.

2. Which describes $h$ on the interval $(4, 6)$ ?

$h$ is increasing and concave down.
$h$ is increasing and concave up.
$h$ is decreasing and concave down.
$h$ is decreasing and concave up.

Show the answer

Two separate questions: is the output rising or falling (increasing), and is the rate of change rising or falling (concavity)? On $(4, 6)$ the depth rises, but it rises more slowly each month, so the graph is concave down.

Correct: $h$ is increasing and concave down.: Right: the depth still rises, but the graph bends downward, so the rate of change of $h$ is decreasing.
$h$ is increasing and concave up.: On $(0, 4)$ the graph bends upward; after $t = 4$ it bends downward.
$h$ is decreasing and concave down.: The depth goes from $6$ to $10$ on this interval, so $h$ is increasing.
$h$ is decreasing and concave up.: That describes $(0, 2)$ , where the depth falls while the graph bends upward.

3. On the interval $(0, 4)$ , how does the rate of change of $h$ behave?

It is decreasing, because $h$ falls and then rises.
It is negative throughout, because $h(4) < h(0)$ .
It stays the same, because the graph is one smooth curve.
It is increasing, so the graph of $h$ is concave up.

Show the answer

Concave up means the rate of change is increasing. Near $t = 0$ the depth falls quickly, by $t = 2$ it has stopped falling, and after that it rises faster and faster until $t = 4$ .

It is decreasing, because $h$ falls and then rises.: The direction of $h$ is a different question. Ask how the rate changes: it goes from strongly negative to positive.
It is negative throughout, because $h(4) < h(0)$ .: On $(2, 4)$ the depth rises, so the rate of change is positive there.
It stays the same, because the graph is one smooth curve.: A smooth curve can bend; this one bends upward on $(0, 4)$ .
Correct: It is increasing, so the graph of $h$ is concave up.: Right: the depth drops fast near $t = 0$ , then more slowly, then starts rising: the rate of change keeps increasing.

Table

A cooling cup of coffee

Temperature T(t) of a cup of coffee, in degrees Fahrenheit, t minutes after it is poured
$t$ (minutes)	0	2	4	6	8
$T(t)$ (degrees Fahrenheit)	180	158	141	128	119

4. What are the changes in $T$ over the four consecutive $2$ -minute intervals in the table?

$-22$ , $-17$ , $-13$ , $-9$
$22$ , $17$ , $13$ , $9$
$-11$ , $-8.5$ , $-6.5$ , $-4.5$
$-22$ , $-39$ , $-52$ , $-61$

Show the answer

$158 - 180 = -22$ , $141 - 158 = -17$ , $128 - 141 = -13$ , $119 - 128 = -9$ .

Correct: $-22$ , $-17$ , $-13$ , $-9$ : Right: each is the later temperature minus the earlier one.
$22$ , $17$ , $13$ , $9$ : These are the sizes of the drops. The temperature falls, so the changes are negative.
$-11$ , $-8.5$ , $-6.5$ , $-4.5$ : These divide each change by $2$ minutes; the question asks for the changes in temperature.
$-22$ , $-39$ , $-52$ , $-61$ : These are changes from $t = 0$ , not over consecutive intervals.

5. Based on the table, which statement about $T$ on $[0, 8]$ is true?

$T$ is decreasing and its rate of change is decreasing.
$T$ is increasing and its rate of change is decreasing.
$T$ is decreasing and its rate of change is increasing.
$T$ is decreasing at a constant rate.

Show the answer

$T$ falls, so it is decreasing. Over equal $2$ -minute steps the changes are $-22, -17, -13, -9$ , which increase, so the rate of change of $T$ is increasing and the graph of $T$ is concave up.

$T$ is decreasing and its rate of change is decreasing.: The changes $-22, -17, -13, -9$ go up, so the rate of change is increasing.
$T$ is increasing and its rate of change is decreasing.: Each temperature is lower than the one before, so $T$ is decreasing.
Correct: $T$ is decreasing and its rate of change is increasing.: Right: the temperature falls, and the changes over equal intervals rise from $-22$ to $-9$ .
$T$ is decreasing at a constant rate.: The changes over equal $2$ -minute intervals are not equal, so the rate is not constant.

6. A car starts from rest and speeds up steadily for $10$ seconds. Let $d(t)$ be the distance it has traveled after $t$ seconds. Which describes the graph of $d$ on $[0, 10]$ ?

Increasing and concave down
Increasing and concave up
Decreasing and concave up
Increasing with a constant rate of change

Show the answer

Distance traveled rises, so $d$ is increasing. Speeding up means the distance covered in each second grows: the rate of change of $d$ is increasing, so $d$ is concave up.

Increasing and concave down: Concave down would mean the distance grows more slowly each second, as when a car slows down.
Correct: Increasing and concave up: Right: the distance keeps growing, and it grows by more each second because the car speeds up.
Decreasing and concave up: Distance traveled cannot go down while the car moves forward.
Increasing with a constant rate of change: A constant rate would mean a constant speed; this car speeds up.

7. A taxi ride costs $C$ dollars for a trip of $m$ miles, and $C$ depends on $m$ . Which is the independent variable?

$m$ , the number of miles
$C$ , the cost
The rate the taxi charges per mile
The time of day

Show the answer

The independent variable is the input, the one that varies freely; the dependent variable is the output that depends on it. Here $C$ is a function of $m$ .

Correct: $m$ , the number of miles: Right: you choose the miles, and the cost follows from them.
$C$ , the cost: The cost depends on the miles, so $C$ is the dependent variable.
The rate the taxi charges per mile: That is a fixed number in the rule, not a variable that changes.
The time of day: Nothing in the description makes the cost depend on the time.

Part 9 · Summary

Summary

Two quantities that change together are described by direction and by how fast. A function is increasing on an interval when larger inputs give larger outputs, decreasing when they give smaller ones. Its rate of change says how fast the output moves; when the rate of change increases the graph is concave up, and when it decreases the graph is concave down.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic