Unit 1 · Topic 1.1 · about 30 minutes · 7 parts · 10 scored checks, 2 of them multiple choice

Change in Tandem

Built from College Board's course framework · Every answer key independently checked

Graph of a cubic function that rises, falls and rises again. It crosses the x-axis at three zeros, x = -3, x = 1 and x = 4. The left part, which bends downward, is labeled concave down, and the right part, which bends upward, is labeled concave up.

Part 1 Think first

Filling a vase

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

Water pours at a steady rate into a flower vase that is narrow at the bottom and wide at the top. As the vase fills, how does the height of the water change?

In short. Two quantities change in tandem when a function rule ties them together, giving each input value exactly one output value.

The domain is the set of inputs and the range is the set of outputs. An input's image is its one output; an output's preimage is every input that gives it.
A function is increasing on an interval if larger inputs always give larger outputs, and decreasing if they always give smaller ones.
A graph is concave up where the rate of change is increasing and concave down where it is decreasing. Zeros are the inputs where the output is 0.

Part 2 Learn it

Four short ideas, each checked as you go

2a. What a function is

A function is a rule that takes each input value to exactly one output value. The set of input values is the domain, and the set of output values is the range. The input variable is the independent variable. The output variable is the dependent variable, because its value depends on the input.

A parking garage that charges by the hour is a function: each number of hours $h$ gives one charge, $C (h)$ . The rule could be a sign with a table on it, a graph, a formula or a sentence. All of them say how input and output vary in tandem.

The output for one input is that input's image: $C (3)$ is the image of 3. Going the other way, the preimage of an output is the set of every input that produces it. If the garage charges 12 dollars for any stay from 2 to 4 hours, the preimage of 12 holds all of those times, not one.

Two functions $f$ and $g$ are equal only when they have the same domain and $f (a) = g (a)$ for every input $a$ in it.

Quick check · Match

No calculator. A garage charges $C (h)$ dollars to park for $h$ hours. Tap a term, then the phrase it matches.

2b. Increasing and decreasing

A function is increasing on an interval if, as the input values increase, the output values always increase. In symbols: for all $a$ and $b$ in the interval, if $a < b$ , then $f (a) < f (b)$ .

A function is decreasing on an interval if, as the input values increase, the output values always decrease: if $a < b$ , then $f (a) > f (b)$ .

The definition says always, so one drop anywhere in the interval rules out increasing. Here is the temperature in a greenhouse, $T (t)$ , in degrees Fahrenheit, $t$ hours after 6 a.m.

Greenhouse temperature
$t$ (hours)	$T (t)$ (degrees)
0	58
2	66
4	79
6	84
8	77
10	70

From $t = 0$ to $t = 6$ , each later reading is warmer, so the data show $T$ increasing. From $t = 6$ to $t = 10$ it is decreasing. Over the whole day, from 0 to 10, $T$ is neither, even though $T (10)$ is bigger than $T (0)$ : the readings fall between 6 and 10, and that fall is what rules out increasing.

Quick check · Multiple choice

No calculator. A function $f$ has $f (1) = 4$ , $f (2) = 7$ , $f (3) = 9$ , $f (4) = 6$ and $f (5) = 2$ . Between these inputs its graph is made of straight segments. On which interval is $f$ decreasing?

2c. Graphs and zeros

The graph of a function is its input-output pairs drawn as points: $(x, y)$ is on the graph when $y = f (x)$ . Read from left to right, the graph shows how the outputs change as the inputs increase. It rises where the function is increasing and falls where it is decreasing.

Where the graph meets the $x$ -axis, the output value is zero. The input values where that happens are the zeros of the function.

In a context, a zero usually marks a moment worth knowing. A phone at 80 percent charge that loses 5 percentage points an hour has battery level $B (t) = 80 - 5 t$ . Setting $B (t) = 0$ gives $t = 16$ : the zero is the hour the battery runs out.

Quick check · Calculate

No calculator. A stone is dropped from a bridge 144 feet above a river. Its height above the water after $t$ seconds is $h (t) = 144 - 16 t^{2}$ feet. What is the zero of $h$ in this context, in seconds?

Your answer seconds

2d. Concave up and concave down

Two graphs can both rise and still look different. One climbs more and more steeply; the other climbs less and less steeply. The difference is in the rate of change, how fast the output changes as the input increases.

Where the rate of change is increasing, the graph is concave up. It bends upward, like a bowl.
Where the rate of change is decreasing, the graph is concave down. It bends downward, like a dome.

This works for falling graphs too. A cup of coffee cools fast at first and then more slowly. Its temperature is decreasing, but its rate of change moves from about $- 6$ degrees per minute toward $- 1$ degree per minute. Those rates are getting larger, so the graph is concave up.

The vase from the start is concave down: the water keeps rising, but by less each second.

Quick check · Sort it

No calculator. Tap a description, then tap the shape of its graph.

Concave up

Concave down

Part 3 Watch it done

Describing a road into a desert basin

Watch each step, then do the last one yourself.

A road runs from a mountain pass down into a desert basin and up the far side. $E (d)$ is the road's elevation in meters compared with sea level, $d$ kilometers from the pass. Between the listed points the road runs straight.

$E (0) = 360$ , $E (4) = 120$ , $E (8) = 0$ , $E (12) = - 80$ , $E (16) = 0$ , $E (20) = 90$ .

Describe how $E$ and $d$ vary together, and find every zero.

Step 1 · Where it decreases
As $d$ goes from 0 to 12, each output is lower than the one before: 360, 120, 0, $- 80$ . So $E$ is decreasing from $d = 0$ to $d = 12$ .
Step 2 · Where it increases
From $d = 12$ to $d = 20$ the outputs rise: $- 80$ , 0, 90. So $E$ is increasing there. The lowest point in the data is 80 meters below sea level, at $d = 12$ .
Why this step?
Why is distance the independent variable here, and not elevation?
Each distance along the road has exactly one elevation, so elevation is a function of distance. The other way round fails: the elevation 0 belongs to two different distances, so its preimage has two inputs and distance cannot be a function of elevation.
Step 3 · The first zero
The output is 0 at $d = 8$ . The road crosses sea level 8 kilometers from the pass, on the way down.
Your step · Calculate

No calculator. Finish the description. At what other distance, in kilometers, is $E (d) = 0$ ?

Your answer km

Part 4 Try it

From a story to a graph

A graph can be built from words alone. Decide what the input and the output are, then ask, part by part, whether the output is rising, falling or holding still, and how fast.

Try it · Put it in order

No calculator. Maya walks from home to a bus stop at a steady pace and waits there a few minutes. She rides the bus farther from home at a much faster steady speed, then gets off and walks a short way back toward home to the library. A graph shows her distance from home against time. Tap its four pieces in order, left to right.

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page.

Free response is 37.5% of the exam score. This one is shaped like Question 1, Function Concepts.

Free-response practice · Question 1 style

Graphing calculator required. A function $g$ has $g (0) = 2$ , $g (1) = 5$ , $g (2) = 3$ and $g (3) = 8$ . A classmate says $g$ is increasing from $x = 0$ to $x = 3$ because $g (3) > g (0)$ . Explain why that reasoning does not show that $g$ is increasing, and use the values to show that it is not.

Your answer stays on this page. It is never sent or saved.

Model answer

Increasing means that for every pair of inputs $a < b$ in the interval, $g (a) < g (b)$ . Comparing the two endpoints checks only one pair. The values give a pair that fails: $1 < 2$ , but $g (1) = 5$ and $g (2) = 3$ , so the output went down while the input went up. So $g$ is not increasing from $x = 0$ to $x = 3$ .

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

States that increasing needs every pair of inputs $a < b$ to give $g (a) < g (b)$ , not only the endpoints.
Names a specific pair from the values where the output goes down, such as $g (1) = 5$ and $g (2) = 3$ .
Concludes that $g$ is not increasing on the interval.

Part 6 Check yourself

A reservoir after a dry summer

The two case questions use the same data. Answer them from the table.

A town measures its reservoir against the normal level once a month. $L (m)$ is the water level in feet above normal, $m$ months after June 1. A negative value means the water is below normal. Between readings the level changes steadily.

Reservoir level compared with normal
$m$ (months)	$L (m)$ (feet)
0	2
1	0
2	−3
3	−4
4	−1
5	0
6	3

Case question 1 · Multiple choice

No calculator. On which of these intervals is $L$ decreasing?

Case question 2 · Calculate

No calculator. How many zeros does $L$ have from $m = 0$ to $m = 6$ ?

Your answer

The rest stand on their own and do not use the case.

Check yourself · Decide and justify

No calculator. Let $f (x) = x$ for every real number $x$ , and let $g (x) = \frac{x^{2}}{x}$ . Are $f$ and $g$ equal functions?

Which reason supports your answer?

Check yourself · Fill the statement

No calculator. A function $s$ gives a student's score on each of five quizzes. Quizzes 1 to 5 scored 8, 10, 7, 10 and 9. Tap a blank, then tap an answer.

The image of 3 is . The preimage of 10 is .

Part 7 Keep it

Hold on to it

This is the first lesson of the course, so there is nothing earlier to bring back yet. From the next lesson on, two questions from earlier lessons appear here.

In three lines

A function gives each input exactly one output. The domain is the set of inputs, the range is the set of outputs, and inputs and outputs vary in tandem.
Increasing means larger inputs always give larger outputs on the interval; decreasing means they always give smaller ones.
Concave up means the rate of change is increasing, concave down means it is decreasing, and zeros are where the output is 0.

Vocabulary

Function: A rule that maps each input value to exactly one output value.
Domain: The set of input values of a function.
Range: The set of output values of a function.
Independent variable: The variable that stands for the input values.
Dependent variable: The variable that stands for the output values, which depend on the input.
Image: The single output value a function gives for one input value.
Preimage: The set of all input values that give a particular output value.
Increasing: On an interval, larger inputs always give larger outputs: if $a < b$ , then $f (a) < f (b)$ .
Decreasing: On an interval, larger inputs always give smaller outputs: if $a < b$ , then $f (a) > f (b)$ .
Concave up: The shape of a graph on an interval where its rate of change is increasing.
Concave down: The shape of a graph on an interval where its rate of change is decreasing.
Zero: An input value where the output is zero; the graph meets the $x$ -axis there.

Course alignment, for teachers

AP Precalculus topic 1.1, Unit 1: Polynomial and Rational Functions.

Skill 2.B: Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology.
Skill 3.A: Describe the characteristics of a function with varying levels of precision, depending on the function representation and available mathematical tools.

This lesson maps all 9 essential knowledge statements of the topic to the part that teaches each one: 9 are checked by a scored question. The map is in the lesson file.