Every concept is classified by what a picture can honestly do for it. The counts
below are computed from that classification, and the build refuses to run if it names a concept
the spine does not contain.
Verdict
Concepts
Share
What it means
direct
467
69%
The concept is a picture. Drawing it is the explanation.
frames
83
12%
A process, not an object. It needs a sequence, which is what the labs in the book are for.
instance
79
12%
Only a particular case can be drawn. The picture is an example of the idea, never the idea.
none
52
8%
No picture is possible. It has to be argued in words or symbols.
629 of 681 concepts can be shown by some picture, and
467 of those are a picture in themselves. The remaining 52 cannot be
drawn at all, and 48% of those
sit in the last stage, which is the argument for having a last stage.
1
The plane itself
44 concepts · 44 drawable
By the endName a position, and say what the two numbers mean.
1
pointA single place, with no width and no height. Everything else on this sheet is built from these.
2
positionWhere something is. On a plane it takes two measurements to pin down, and neither alone is enough.
3
locationThe pair of numbers that names a position. Give them to anyone with the same axes and they will find the same spot.
4
number lineA line with the numbers laid along it in order. One measurement fixes a place, because there is only one way to move.
5
one dimensionOnly one way to move: forward or back. That is why a single number is enough to say where you are.
6
two dimensionsTwo independent ways to move, so it takes two numbers. Change one and the point slides; change both and it goes anywhere.
7
originThe point both measurements are taken from, written (0, 0). Nothing about it is special except that everyone agrees on it.
8
axisOne of the two lines everything is measured against. Each carries its own number line.
9
x-axisThe line running across, where the second measurement is zero. Every point on it has the form (something, 0).
10
y-axisThe line running up, where the first measurement is zero. Every point on it has the form (0, something).
11
axesThe two lines together. They cross at the origin and cut the plane into four regions.
12
coordinate planeThe whole flat sheet, with axes and a grid, on which any pair of numbers has a home.
13
Cartesian planeThe same sheet, named after Descartes, who joined algebra to geometry by giving every point a pair of numbers.
14
ordered pairTwo numbers in a fixed order, written in brackets. The order is the whole point: swap them and you have a different place.
15
coordinateOne of the two numbers. On its own it narrows a point to a line, not to a place.
16
x-coordinateHow far across, measured from the y-axis. Positive to the right, negative to the left.
17
y-coordinateHow far up, measured from the x-axis. Positive above, negative below.
18
abscissaThe older name for the x-coordinate. You will meet it in books rather than in classrooms.
19
ordinateThe older name for the y-coordinate, and the one that gave "ordered pair" its name.
20
plotting a pointTurning a pair of numbers into a place: go across by the first, then up by the second. Always in that order.
21
reading a pointThe reverse: drop from a marked place to each axis and read the two numbers off.
22
quadrantOne of the four regions the axes cut the plane into. Which one a point is in is decided entirely by the signs of its two numbers.
23
first quadrantTop right, where both numbers are positive. Most graphs of real quantities live here, because lengths and times are rarely negative.
24
second quadrantTop left: x negative, y positive.
25
third quadrantBottom left, where both numbers are negative.
26
fourth quadrantBottom right: x positive, y negative.
27
sign of a coordinateWhether each number is positive or negative. The two signs together say which quadrant you are in, without measuring anything.
28
positive directionRight along the x-axis and up the y-axis. A convention, agreed rather than discovered, but agreed everywhere.
29
negative directionLeft and down. Not smaller or lesser, just the opposite way.
30
gridlineA line of the grid. Every point on a vertical one shares an x-coordinate, which is what makes it useful for reading a graph.
31
lattice pointA point where both numbers are whole. Easy to plot exactly, which is why tables of values tend to use them.
32
tick markThe small strokes counting units along an axis. Without them the picture has shape but no size.
33
scaleHow much one step of the grid is worth. Change it and the same relationship can look steep or nearly flat.
34
unitOne step. Everything measured on the plane is counted in these, and the answer means nothing until you say what one is.
35
equal scalingBoth axes using the same size of step. Only then does a square look square and an angle look like itself.
36
distorted scalingAxes with different steps. Legal and often useful, but shape can no longer be trusted: circles turn into ovals and steepness lies.
37
axis labelThe name of what each axis measures. Without it a graph is a shape with no subject.
38
the point (0, 0)The origin written as a pair. It is an ordinary point that happens to be where both measurements read nothing.
39
distance from an axisThe shortest way to an axis, which is straight at it. That distance is exactly what the other coordinate measures.
40
above and belowWhich side of the x-axis a point is on, which is just the sign of its second number.
41
left and rightWhich side of the y-axis a point is on, which is just the sign of its first number.
42
reference frameThe origin and axes you have agreed to measure from. Move them and every coordinate changes, though nothing has actually moved.
43
convention (input first)The agreement that the first number is measured across. Nothing forces it; everyone simply does the same thing so the numbers travel.
44
the pair (3, 1) against (1, 3)Two different points from the same two numbers. This is the clearest reason the order has to be fixed.
2
Straight lines and segments
50 concepts · 50 drawable
By the endMeasure a line: how long, how steep, where it crosses.
45
straight lineThe path that never turns. It carries on past the edge of any picture of it, which is why a drawn line is always a sample of one.
46
line segmentA piece of a line with two ends. Unlike a line it has a length you can measure.
47
rayHalf a line: one end, and no other. A beam of light is the usual picture.
48
endpointWhere a segment stops. Two of them, and between them everything belongs to the segment.
49
interior pointA point on a segment that is not an end. Between, strictly.
50
betweennessOne point lying on the segment joining two others. It is the idea that lets a line be ordered.
51
length of a segmentHow far apart the two ends are. Drop the horizontal and vertical legs and it is the hypotenuse of a right triangle.
52
distance formulaPythagoras written for two coordinate pairs: square the two differences, add, take the root.
53
Pythagoras in the planeThe rule that makes distance measurable at all. Every length on this sheet is an application of it.
54
midpointThe point exactly halfway. Its two coordinates are the averages of the two ends.
55
midpoint formulaAverage each coordinate separately. There is nothing more to it than that.
56
section formulaThe midpoint generalised: a point dividing a segment in any stated ratio, not just in half.
57
internal divisionA dividing point that lies inside the segment. Both parts point the same way.
58
external divisionA dividing point beyond one end. The ratio is signed, and one part now runs backwards.
59
ratio along a segmentHow the two parts compare. It fixes a point without ever mentioning a length.
60
collinear pointsPoints that all sit on one line. Three points are collinear when the slope between any two of them is the same.
61
horizontal lineA line of constant height. Its equation names only y, because x is free to be anything.
62
vertical lineA line of constant position across. Its equation names only x, and it is the one kind of line that is not a function.
63
oblique lineAny line that is neither flat nor upright. Both coordinates change as you move along it.
64
slopeHow much the line climbs for each step across. One number that describes the whole line, because a line never changes its mind.
65
gradientThe same number as slope, under the name used in British schools and in physics.
66
riseThe vertical part of a step along the line. Negative if the line is going down.
67
runThe horizontal part of the same step. Usually taken as positive, so the sign of the answer comes from the rise.
68
rise over runDivide one by the other and the size of the step drops out. That is why any two points on a line give the same slope.
69
steepnessHow sharply a line climbs. The bigger the slope in size, the steeper, whichever way it leans.
70
positive slopeThe line climbs left to right. As the input grows, so does the output.
71
negative slopeThe line falls left to right. As the input grows, the output shrinks.
72
zero slopeNo climb at all. Move as far across as you like and the height never changes.
73
undefined slopeAn upright line has no run to divide by, and dividing by nothing is not an answer. That is why it is undefined rather than infinite.
74
angle of inclinationThe angle the line makes with the horizontal. Slope and angle carry the same information in different units.
75
parallel linesSame slope, different position. They never meet, however far either is extended.
76
perpendicular linesCrossing at a right angle. Their slopes multiply to −1, which is the algebraic form of that fact.
77
negative reciprocalFlip the fraction and change the sign. Doing that to a slope turns a line through a right angle.
78
x-interceptWhere the line crosses the horizontal axis, so the output is zero. Solving a linear equation is exactly finding this.
79
y-interceptWhere the line crosses the vertical axis, so the input is zero. It is the starting value before anything happens.
80
equation of a lineThe condition every point on the line satisfies, and no other point does. The line and the equation are the same object.
81
linear equationAn equation where the letters appear only to the first power. That restriction is exactly what makes the graph straight.
82
slope-intercept formy = mx + c. The two numbers you can read straight off a graph: how steep, and where it starts.
83
point-slope formBuilt from one point and a slope. Useful when you know where a line goes and how steep it is, but not where it crosses.
84
two-point formTwo points are enough to fix a line, because they fix both the slope and the position.
85
standard formBoth letters on one side, as Ax + By = C. It treats x and y evenly, which suits solving pairs of equations.
86
general formEverything moved to one side, equal to zero. It is the form a computer prefers, because there is nothing to rearrange.
87
intercept formWritten from the two lengths the line cuts off the axes. It says at a glance where the line meets each one.
88
distance from a point to a lineThe shortest route, which is always the perpendicular one. Any other path to the line is longer.
89
foot of the perpendicularWhere that shortest route lands. It is the point on the line closest to P.
90
angle between two linesHow far one must turn to lie along the other. It depends only on their slopes, not on where they cross.
91
perpendicular bisectorThe line cutting a segment in half at a right angle. Every point on it is the same distance from both ends.
92
locus of equidistant pointsThe set of all points equally far from two others. It turns out to be exactly that perpendicular bisector.
93
direction of travelA segment with an arrow on it. Adding direction turns a length into the beginning of a vector.
94
a line as a set of pointsNot a stroke of ink, but every point whose coordinates satisfy the equation. The drawn line is just the ones that fit on the page.
3
Regions, shapes and systems
52 concepts · 52 drawable
By the endTurn a picture of a region into arithmetic, and back.
95
polygonA closed shape made only of straight sides. Add a side and you have a new one; there is no upper limit.
96
vertexA corner, where two sides meet. A polygon has exactly as many corners as sides.
97
sideOne straight edge. Going round them all in order is what closes the shape.
98
triangleThree points not in a line, joined up. The simplest polygon, and the one every other polygon can be cut into.
99
right triangleOne corner square. This is the shape Pythagoras applies to, which is why every distance on this sheet is one.
100
isosceles triangleTwo sides equal, and with them two equal angles. Symmetry about a line through the odd corner.
101
equilateral triangleAll three sides equal, so all three angles are sixty degrees. The most symmetric triangle there is.
102
quadrilateralFour straight sides. Every rectangle, square, rhombus and trapezium below is one of these with something extra promised.
103
rectangleFour right angles. Opposite sides come out equal as a consequence, not as a second condition.
104
squareA rectangle whose sides are all equal. It is the unit of area: everything else is measured in these.
105
parallelogramBoth pairs of opposite sides parallel. Push a rectangle sideways and you have one, with the same area.
106
rhombusA parallelogram with all four sides equal. Its diagonals cross at right angles.
107
trapeziumExactly one pair of parallel sides. Its area is the average of those two, times the distance between them.
108
perimeterThe distance all the way round. Add the sides; there is no formula beyond that.
109
area of a triangleHalf the base times the height. Slide the top corner along a parallel line and the area never changes, because neither does.
110
shoelace formulaArea straight from the corner coordinates, with no height to find. Multiply crosswise round the shape and subtract.
111
determinant formThe same computation written as a determinant. It is why area and determinants turn out to be the same idea.
112
area of a polygonCut it into triangles from one corner and add them. Any polygon can be cut this way, so nothing else is needed.
113
regionA part of the plane rather than a line through it. Regions are what inequalities describe.
114
half-planeEverything on one side of a line. Two of them, and the line between, make up the whole plane.
115
inequalityA condition met by a whole region rather than by a single answer. Its picture is shading, not a point.
116
linear inequalityThe boundary is a straight line, so the region is a half-plane. Test any single point to find out which side is wanted.
117
strict inequalityThe boundary itself is excluded, drawn dashed. The difference matters at the edge and nowhere else.
118
boundary lineThe edge of a region. Solid when it belongs to the region, dashed when it does not.
119
shaded regionWhere two or more conditions hold at once. Each one alone allows more; together they allow less.
120
feasible regionEverything allowed by every constraint. Add a constraint and it can only shrink.
121
convex regionJoin any two of its points and the segment stays inside. Feasible regions from linear constraints are always like this.
122
bounded regionIt fits inside some box. A bounded region always has a highest and a lowest point of anything measured on it.
123
unbounded regionIt runs on without limit. A quantity measured over one may have no largest value at all.
124
intersection of two linesWhere both lines pass at once. Unless they are parallel, there is exactly one such place.
125
point of intersectionThe one point satisfying both equations. Reading off its coordinates is solving the pair.
126
simultaneous equationsTwo conditions to hold at the same time. Each is a line; the answer is where they meet.
127
system of equationsAny number of equations considered together. More equations usually means fewer answers, and often none.
128
unique solutionDifferent slopes, so the lines must cross, and can only cross once.
129
no solutionSame slope, different intercept. Parallel lines never meet, so no pair of numbers satisfies both.
130
infinitely many solutionsThe two equations describe the same line, so every point on it solves both. Usually one is a multiple of the other.
131
consistent systemIt has at least one solution. Either the lines cross, or they coincide.
132
inconsistent systemNo solution at all. The conditions contradict each other, and the algebra ends in something false like 0 = 5.
133
dependent equationsOne equation carries no new information, because it is a rescaling of another. Two equations, one condition.
134
graphical solutionDraw both and look. It gives an answer you can see and trust to about a grid square, which is often enough.
135
substitution methodUse one equation to express a letter, then put it into the other. Geometrically, you have fixed one coordinate first.
136
elimination methodAdd or subtract the equations so one letter cancels. The combination is a third line through the same crossing.
137
linear programmingFind the best allowed value of something. Because the region is convex and the objective is straight, the answer is always at a corner.
138
objective functionThe quantity being maximised or minimised. Its lines of equal value are parallel, so sliding them across the region finds the best point.
139
corner pointA vertex of the feasible region. Checking only these is enough, which turns an infinite search into a short list.
140
concurrencyThree or more lines through one point. It is not automatic, which is why it is worth a name when it happens.
141
mediansEach line from a corner to the midpoint of the opposite side. All three always pass through one point.
142
centroidWhere the medians meet, and the average of the three corners. A cardboard triangle balances on it.
143
circumcentreThe centre of the circle through all three corners. It is equally far from each, so it sits on every perpendicular bisector.
144
incentreThe centre of the largest circle that fits inside. It is equally far from each side, rather than from each corner.
145
orthocentreWhere the three altitudes meet. Unlike the others it can fall outside the triangle entirely.
146
Euler lineThe circumcentre, centroid and orthocentre always lie on one line, in that order, whatever the triangle. A surprise that has no obvious reason to be true.
4
The function idea
50 concepts · 47 drawable
By the endSay what a rule promises, and test whether a picture keeps it.
147
relationAny pairing at all between two sets. No promises: an input may have many partners, or none.
148
functionA relation that promises exactly one output for every allowed input. That single promise is the whole definition.
149
ruleThe instruction that turns an input into an output. It can be words, a formula, a table, or a machine.
150
inputWhat goes in. Choosing it is the one free act; everything after is decided by the rule.
151
outputWhat comes out. You do not choose it; the input and the rule between them settle it.
152
domainEvery input the rule will accept. Stating it is part of stating the function, not an afterthought.
153
rangeEvery output the rule actually produces. Not what it might produce: what it does.
154
codomainThe set outputs are allowed to come from. It may be bigger than the range, and often is, because it is declared rather than computed.
155
imageWhere a particular input is sent. The image of 3 under squaring is 9.
156
preimageWhich input produced a given output. Running the arrow backwards, which may find one answer, several, or none.
157
mappingAnother word for a function, used when the sets are not numbers. It emphasises the sending rather than the calculating.
158
arrow diagramTwo columns and arrows between them. The clearest way to see whether a rule keeps its promise.
159
set of ordered pairsA function written out as a list of what goes to what. Nothing is hidden: the list is the whole rule.
160
well-definedThe rule settles on one answer, and the same answer every time. Without this, nothing built on top of it can be trusted.
161
one input one outputThe promise, stated as plainly as it can be. Two inputs may share an output; one input may not split.
162
the forkOne input with two arrows. It is the only way a relation fails to be a function, and it is always fatal.
163
vertical line testSweep an upright line across a graph. It works because a vertical line collects every output at one input.
164
horizontal line testSweep a flat line instead. Two hits means two inputs share an output, so the rule cannot be reversed.
165
one-to-oneNo output is used twice. It is the condition that lets a rule be run backwards.
166
injectiveThe formal name for one-to-one. Every output is used at most once; some may go unused.
167
many-to-oneSeveral inputs landing on the same output. Perfectly legal, and what squaring does to 3 and −3.
168
ontoNothing in the target is left out. Every possible output is actually reached by something.
169
surjectiveThe formal name for onto. Nothing in the codomain is wasted.
170
bijectiveBoth at once: nothing shared and nothing left out. A perfect pairing, and exactly what an inverse needs.
171
function notationA compact way to name the output a rule gives an input. The brackets mean "apply", never "multiply".
172
f(x)Read "f of x". The letter names the rule, the bracket holds whatever it is being applied to.
173
evaluatingInput given, output wanted. Substitute and compute; there is only ever one answer.
174
solving for the inputOutput given, input wanted. The arrow runs backwards, and there may be more than one answer, or none.
175
independent variableThe one you set. It goes across, and nothing in the rule constrains it beyond the domain.
176
dependent variableThe one that follows. It goes up, and you never choose it directly: the rule and your input decide it.
177
argumentWhatever sits inside the brackets. It need not be a number: an expression works the same way.
178
value of a functionThe number that comes out at a stated input. One input, one value, always.
179
substituting into a ruleWhatever arrives goes into every blank. Reading the rule with a blank in it makes the next step obvious.
180
table of valuesA few rows of the rule, written out. It cannot show every input, but it can show the pattern.
181
graph of a functionEvery input-output pair plotted at once. The table is a sample of it; the graph is all of it.
182
four representationsWords, table, formula and graph. Fluency is moving between them in any direction, not reciting each in turn.
183
natural domainEverything the formula will take, before anyone restricts it further. Found by scanning for what would break.
184
implied domainThe domain a formula states without anyone writing it down. Silence is not permission: it is the formula speaking.
185
contextual domainNarrowed by what the quantity means rather than by the algebra. A count cannot be negative even when the formula would allow it.
186
restricted domainInputs deliberately withheld. It is how a rule that cannot be reversed is made reversible.
187
excluded valueA single input the rule refuses. Drawn as a hollow endpoint, or as a hole in a curve.
188
division by zeroThe commonest reason an input is refused. It is not that the answer is large: there is no answer.
189
even root of a negativeThe other common refusal. No real number squares to something negative, so the root has nothing to return.
190
interval notationA stretch of the line written in brackets. Square includes the end, round excludes it.
191
open intervalNeither end included. Every point in it has room on both sides, which is what makes it useful for limits.
192
closed intervalBoth ends included. A continuous rule on one of these always attains a highest and a lowest value.
193
half-open intervalOne end in, one out. Common when a quantity starts somewhere definite and runs on without reaching a limit.
194
union of intervalsTwo or more stretches taken together. Removing a point from the line always produces one.
195
unbounded intervalOne end and no other. Infinity always takes a round bracket, because it is a direction rather than a place you arrive at.
196
set-builder notationA set written as the condition its members satisfy, rather than as a list. It works when the list would be infinite.
5
Families of curves
67 concepts · 65 drawable
By the endRecognise a shape from its rule, and a rule from its shape.
197
constant functionThe same output whatever goes in. Flat, and the only rule whose graph a horizontal line test fails everywhere.
198
identity functionGives back exactly what it was handed. It is the diagonal, and the mirror every inverse is reflected in.
199
linear functionOne steepness, everywhere. Equal steps in give equal steps out, which is what makes it the simplest kind of change.
200
affine functionA line that need not pass through the origin. Strictly, only lines through the origin are linear; the rest are affine.
201
quadratic functionA squared term is enough to bend it. One turning point, and a mirror through it.
202
parabolaThe curve a quadratic draws. Every point on it is equally far from a fixed point and a fixed line, which is the other way to define it.
203
vertex of a parabolaThe single turning point, and the only place the curve is flat. Everything about a quadratic is easiest to read from here.
204
axis of symmetryThe vertical line the curve folds onto itself along. It always runs through the vertex.
205
completing the squareRewriting a quadratic so the vertex can be read straight off. The curve does not change; only the way it is written does.
206
vertex forma(x − h)² + k. The two numbers h and k are exactly where the vertex is, so no work is needed to find it.
207
factored formWritten as a product, so the roots are visible. Each bracket vanishing gives one crossing.
208
rootAn input that makes the output zero. On a graph it is a crossing of the horizontal axis.
209
zero of a functionThe same thing as a root, under the name used when the rule is not a polynomial.
210
discriminantOne number that says how many roots there are before you find any. Positive gives two, zero gives one, negative gives none.
211
repeated rootThe curve touches the axis and turns back instead of crossing. Two roots that have landed on the same place.
212
cubic functionA third power allows two turns. Both ends head opposite ways, so it always crosses the axis at least once.
213
quartic functionA fourth power allows three turns, and both ends go the same way. That is why it can have a W shape.
214
polynomialSums of whole-number powers. Smooth everywhere, defined everywhere, and with no breaks or corners anywhere.
215
degreeThe highest power present. It caps the number of turns at one less, and decides what the ends do.
216
leading coefficientThe number on the highest power. Its sign alone decides which way the far ends of the curve point.
217
end behaviourWhat happens far from the origin, where the highest power drowns out everything else.
218
turning pointWhere the curve stops rising and starts falling, or the reverse. Momentarily flat.
219
local maximumHigher than everything immediately around it, though possibly not the highest anywhere.
220
local minimumLower than everything immediately around it. The dip a ball would settle in, if the curve were a track.
221
global maximumThe highest the rule ever gets, anywhere in its domain. A rule may have none at all.
222
global minimumThe lowest it ever gets. On a closed interval a continuous rule always has one; on an open one it may not.
223
rational functionOne polynomial divided by another. Wherever the bottom vanishes the rule has nothing to give, so the curve breaks.
224
asymptoteA line the curve gets arbitrarily close to and never reaches. It is a statement about forever, not about the visible part.
225
vertical asymptoteAn upright line the curve runs alongside without limit. It marks an input the rule refuses.
226
horizontal asymptoteA level the curve settles toward far out. It answers what happens in the long run.
227
oblique asymptoteA slanted line approached at both ends. It appears when the top of a fraction outgrows the bottom by exactly one power.
228
hole in a graphA single point missing from an otherwise unbroken curve. It happens when a factor cancels, which is illegal at exactly one input.
229
reciprocal functionOne over the input. Two branches, each hugging both axes, and nothing at all at zero.
230
hyperbola (rectangular)The curve whose two coordinates always multiply to the same number. Double one and the other halves.
231
square root functionUndoes squaring, but only for inputs that are not negative. It starts at the origin and flattens as it climbs.
232
radical functionAny rule with a root in it. Shifting what is under the root moves where the curve is allowed to start.
233
cube root functionUndoes cubing, and unlike the square root it accepts negatives, because cubing keeps the sign.
234
absolute value functionDistance from zero, so the sign is thrown away. Two straight pieces meeting at a corner.
235
piecewise functionDifferent rules on different stretches. One function, described in parts, which is how most real quantities behave.
236
step functionConstant, then it jumps, then constant again. Postage and parking charges work exactly like this.
237
floor functionThe whole number at or below the input. It rounds down, including for negatives, where that surprises people.
238
ceiling functionThe whole number at or above the input. It rounds up, always.
239
signum functionReports only the sign: minus one, nothing, or one. It throws away everything except direction.
240
exponential functionEqual steps in multiply the output by a fixed factor. That is why it eventually outruns any power, however large.
241
baseThe factor each step multiplies by. Bigger than one and it grows; between zero and one and it decays.
242
exponential growthThe amount added depends on how much there already is. Slow at first, and then not.
243
exponential decayA fixed fraction lost each step, so it approaches nothing without ever arriving.
244
doubling timeHow long growth takes to double. For an exponential it is the same however much there already is, which is the surprising part.
245
half-lifeHow long decay takes to halve. Like doubling time, it does not depend on where you start.
246
the number eThe one base whose steepness equals its own height everywhere. About 2.718, and it is not a coincidence that calculus keeps finding it.
247
logarithmAsks what power gives this number. It is the exponential run backwards, so it undoes it.
248
logarithmic functionDefined only for positive inputs, and it climbs ever more slowly. It never stops climbing, which is easy to miss.
249
natural logarithmThe logarithm to base e. It is the one whose slope at x is exactly one over x, which is why calculus prefers it.
250
common logarithmThe logarithm to base ten. It counts digits, which is why it is the one used for scales like pH and decibels.
251
log lawsThey turn multiplying into adding. That is what made logarithms worth inventing, three centuries before calculators.
252
sineThe height of a point going round a circle, plotted against the angle. It repeats forever because the circle does.
253
cosineThe across-ness of the same circling point. The identical wave, started a quarter turn earlier.
254
tangent functionSine divided by cosine. Wherever cosine is zero it has nothing to give, so it breaks and starts again.
255
amplitudeHow far the wave reaches from its centre line. It changes the height and nothing else.
256
periodThe horizontal length of one full repeat. Changing it stretches the wave sideways without changing its height.
257
phase shiftThe wave slid sideways. Nothing about its shape changes; only where it starts.
258
frequency (of a wave)How many repeats fit into a given stretch. It is the period turned upside down.
259
radian measureAngle measured by the arc it cuts, in units of the radius. It is the measure that makes calculus of waves come out clean.
260
degree measureA full turn cut into 360 parts. Older, more familiar, and arbitrary: 360 was chosen because it divides neatly.
261
unit circleThe circle of radius one. Every point on it hands you a cosine and a sine at once, which is where both come from.
262
periodicityRepeating the same shape at fixed intervals, forever. Knowing one cycle is knowing all of them.
263
sinusoidAny wave of this shape, whatever its height, length and starting point. Sound, light and alternating current are all made of these.
6
Moving a curve
31 concepts · 29 drawable
By the endPredict where a graph goes before drawing it.
264
parent functionThe plainest member of a family, before anything is done to it. Every other member is this one moved, stretched or flipped.
265
transformationA rule applied to a whole curve at once. The shape is carried somewhere else rather than redrawn.
266
translationSliding without turning or resizing. Every point moves the same distance in the same direction.
267
horizontal shiftMoving sideways. It is done inside the bracket, and it moves the opposite way to the sign you write.
268
vertical shiftMoving up or down. It is done outside the bracket, and it moves the way the sign says.
269
inside the bracketChanges made to the input, before the rule runs. They act sideways, and they act backwards.
270
outside the bracketChanges made to the answer, after the rule has run. They act vertically, and they behave as written.
271
shift against the signx − 2 sends the curve right, not left. The bracket asks what to feed the rule, so to get the old answer you must arrive two later.
272
reflectionFlipping the curve across a line, as a mirror would. Distances to the mirror are preserved; sides are swapped.
273
reflection in the x-axisNegate the answer, and the curve turns upside down. Anything sitting on the axis does not move.
274
reflection in the y-axisNegate the input, and left swaps with right. For an even curve nothing appears to happen at all.
275
reflection in y = xSwap the two coordinates and the diagonal becomes the mirror. This is exactly what taking an inverse does.
276
stretchPulling the curve away from a line, so distances from it are multiplied. The line itself stays put.
277
compressionThe same operation with a factor under one: everything is pulled toward the line rather than away from it.
278
vertical stretchMultiply the answer and every height scales. It happens outside the bracket, so the factor means what it says.
279
horizontal stretchMultiply the input and the curve squeezes sideways by the reciprocal. Feed it 2x and it finishes in half the distance.
280
scale factorThe number every distance is multiplied by. Above one it grows, below one it shrinks, and negative flips as well as scales.
281
dilationA stretch about a fixed centre. Every point moves along the line joining it to that centre.
282
invariant pointA point a transformation leaves exactly where it was. It is the anchor the rest of the picture turns around.
283
order of transformationsDoing them in a different order gives a different curve. Stretch then shift is not shift then stretch.
284
composition of transformationsSeveral applied one after another, read from the inside out. The written form is a set of instructions in reverse.
285
even functionFeeding it the negative gives back the same answer. Its graph is unchanged by a flip in the vertical axis.
286
odd functionFeeding it the negative flips the answer too. Turning the graph half a turn about the origin leaves it alone.
287
symmetry about the y-axisThe left half is the right half, mirrored. Only even rules do this.
288
symmetry about the originHalf a turn about the origin puts the curve back on itself. Every point has a partner directly opposite.
289
rotational symmetryTurning by a fixed fraction of a full circle leaves the shape looking identical. How many such turns is its order.
290
image curveWhat is left after the transformation has been applied. The original is the object; this is its image.
291
mapping notationWriting where a general point goes, rather than describing the move in words. It leaves no room for ambiguity.
292
transformation matrixA small grid of numbers that performs the move by multiplication. Its two columns are simply where the two unit arrows land.
293
rotation about the originTurning everything by the same angle around a fixed point. Distances from that point never change.
294
shearSliding each row sideways by an amount that grows with its height. A rectangle becomes a leaning one, of exactly the same area.
7
Conics, loci and other coordinates
46 concepts · 45 drawable
By the endDescribe a curve by the condition its points satisfy.
295
locusThe set of every point obeying some condition, and nothing else. Curves stop being shapes you draw and become answers to a question.
296
circleEvery point the same distance from one point. The condition is short enough to check on any point you like.
297
centreThe point every distance is measured from. Move it and the whole circle goes with it, unchanged in size.
298
radiusThe fixed distance in the condition. It is the only thing that decides how big the circle is.
299
equation of a circlePythagoras applied to the condition: the squared distance from the centre equals the squared radius.
300
general form of a circleThe same circle multiplied out, with the centre hidden. Completing the square in both letters puts it back.
301
diameterA chord that passes through the centre, and the longest one there is. It is twice the radius.
302
chordA straight segment joining two points on the circle. It cuts the circle into two pieces.
303
arcA stretch of the circle itself, rather than of the plane inside it. Two points cut off a short one and a long one.
304
sectorThe slice between two radii, arc included. A pie piece, and its area is the fraction of the turn it takes up.
305
segment of a circleThe region a chord cuts off, without the centre. What is left of a sector once the triangle is removed.
306
tangent to a circleA line touching at exactly one point. It always meets the radius there at a right angle, which is the fact everything else is built on.
307
secant to a circleA line cutting the circle twice. Slide it outwards and the two crossings meet; at that moment it is a tangent.
308
point of contactThe single point a tangent shares with the curve. It is where the two crossings of a secant have collapsed together.
309
conic sectionThe curves a flat cut through a cone can produce. Tilt the cut further and one becomes the next.
310
ellipseThe distances to two fixed points always add to the same total. A loop of string round two pins draws one.
311
major axisThe longest way across, through both foci. Half of it is the number that fixes the total distance.
312
minor axisThe shortest way across, at right angles to the major axis. It decides how round or how flat the ellipse looks.
313
focusA point the curve is defined in terms of rather than drawn around. Light from it reflects off the curve in a single direction.
314
fociTwo of them, for an ellipse and for a hyperbola. Slide them together and the ellipse becomes a circle.
315
eccentricityOne number saying how far from circular the curve is. Zero is a circle, below one an ellipse, exactly one a parabola, above one a hyperbola.
316
directrixA fixed line the curve is measured against. Distance to the focus divided by distance to this line is the eccentricity, at every point.
317
parabola as a conicThe single case where the two distances are equal. It is the boundary between the closed curves and the open ones.
318
latus rectumThe chord through the focus, parallel to the directrix. It is a quick measure of how wide the curve opens.
319
hyperbolaThe distances to two fixed points always differ by the same amount. That gives two separate branches, one round each focus.
320
transverse axisThe line through both vertices and both foci. It is the direction the branches open away from.
321
conjugate axisThe perpendicular direction, which the curve never crosses. Its length still sets how steeply the branches spread.
322
asymptotes of a hyperbolaTwo crossing lines the branches settle onto. Far enough out, a hyperbola is indistinguishable from them.
323
degenerate conicWhat the cut gives when it passes through the tip of the cone: a single point, one line, or a crossing pair.
324
parametric equationx and y each given separately in terms of a third quantity. The curve becomes a journey rather than a condition.
325
parameterThe third quantity doing the driving, often standing for time. It appears in the working and not in the picture.
326
parametrisationA choice of how to travel along a curve. The same path can be walked at different speeds, and neither is more correct.
327
eliminating the parameterGetting rid of t to leave a relation between x and y alone. What survives is the shape, with the timing thrown away.
328
polar coordinatesLocating a point by how far and in what direction, instead of by two sideways steps. Anything built around a centre is simpler this way.
329
poleThe point everything is measured from, the polar version of the origin. Every direction starts here.
330
polar axisThe ray angles are measured from, usually pointing right. It plays the part the positive x-axis plays elsewhere.
331
radial coordinateThe distance out from the pole, written r. Changing it alone moves the point straight in or straight out.
332
angular coordinateThe direction, written θ. Changing it alone swings the point round a circle of fixed size.
333
polar to Cartesian conversionx is r cos θ and y is r sin θ. It is one right-angled triangle, with the distance as its hypotenuse.
334
polar curveA rule giving the distance for each direction. Sweep the direction all the way round and the rule draws the shape.
335
circle in polar formA circle round the pole needs only r = a. One passing through the pole needs the angle, and comes out as r = 2a cos θ.
336
rose curvePetals, from r = cos kθ. An odd k gives k petals, an even k gives twice as many, which surprises everyone the first time.
337
cardioidA heart shape, from r = a(1 + cos θ). The distance drops to nothing at one direction, which makes the dimple.
338
limaçonr = b + a cos θ. When b is smaller than a the distance goes negative for a while, and the curve tucks an inner loop inside itself.
339
Archimedean spiralr = aθ, so the distance grows evenly with the angle. Every turn is the same width apart, unlike a shell.
340
lemniscateA figure eight, from r² = a² cos 2θ. It exists only where the right-hand side is positive, so two lobes appear and the rest is empty.
8
Data on the plane
41 concepts · 41 drawable
By the endFit a curve to measurements, and say how well it fits.
341
scatter plot
342
bivariate data
343
explanatory variable
344
response variable
345
trend
346
positive correlation
347
negative correlation
348
no correlation
349
correlation coefficient
350
causation against correlation
351
line of best fit
352
least squares
353
regression line
354
residual
355
residual plot
356
interpolation
357
extrapolation
358
outlier
359
influential point
360
linear model
361
exponential model
362
power model
363
logistic model
364
curve fitting
365
goodness of fit
366
time series
367
histogram
368
frequency (of a value)
369
cumulative frequency
370
box plot
371
quartile
372
median
373
mean
374
standard deviation
375
normal curve
376
area as probability
377
z-score
378
probability density
379
cumulative distribution
380
sampling
381
sample size
9
Change, and the bridge to limits
49 concepts · 47 drawable
By the endSay which two things were subtracted, and what was divided by what.
382
difference
383
delta
384
increment
385
change
386
new minus old
387
signed change
388
rate
389
per unit
390
rate of change
391
average rate of change
392
secant line
393
slope of a secant
394
chord of a curve
395
interval
396
width of an interval
397
difference quotient
398
h notation
399
shrinking the interval
400
instantaneous rate
401
tangent line
402
slope of a tangent
403
point of tangency
404
approaching a value
405
neighbourhood
406
arbitrarily close
407
limit
408
left-hand limit
409
right-hand limit
410
two-sided limit
411
existence of a limit
412
limit against value
413
indeterminate form
414
zero over zero
415
factor and cancel
416
continuity
417
continuous at a point
418
continuous on an interval
419
discontinuity
420
removable discontinuity
421
jump discontinuity
422
infinite discontinuity
423
limit at infinity
424
asymptotic behaviour
425
squeeze theorem
426
intermediate value theorem
427
extreme value theorem
428
epsilon
429
delta (the tolerance)
430
epsilon-delta definition
10
Derivatives
63 concepts · 56 drawable
By the endMeasure how fast, at a point, and use it to find the best of something.
431
derivative
432
differentiation
433
differentiable
434
derivative at a point
435
derivative as a function
436
prime notation
437
Leibniz notation
438
dy/dx
439
operator notation
440
differentiability implies continuity
441
non-differentiable point
442
corner
443
cusp
444
vertical tangent
445
power rule
446
constant rule
447
constant multiple rule
448
sum rule
449
difference rule
450
product rule
451
quotient rule
452
chain rule
453
outer and inner function
454
implicit differentiation
455
logarithmic differentiation
456
derivative of an exponential
457
derivative of a logarithm
458
derivatives of sine and cosine
459
derivative of an inverse function
460
second derivative
461
higher derivatives
462
concavity
463
concave up
464
concave down
465
point of inflection
466
critical point
467
stationary point
468
first derivative test
469
second derivative test
470
increasing function
471
decreasing function
472
monotonic
473
local extremum
474
global extremum
475
optimisation
476
constraint
477
related rates
478
displacement
479
velocity
480
speed
481
acceleration
482
marginal cost
483
elasticity
484
tangent line approximation
485
linearisation
486
differential
487
Newton's method
488
Rolle's theorem
489
mean value theorem
490
L'Hopital's rule
491
Taylor polynomial
492
Maclaurin series
493
radius of convergence
11
Integrals
61 concepts · 53 drawable
By the endAdd up a changing quantity, and know why that undoes a derivative.
494antiderivative
495indefinite integral
496constant of integration
497integrand
498integral sign
499variable of integration
500definite integral
501limits of integration
502area under a curve
503signed area
504net area
505Riemann sum
506left-hand sum
507right-hand sum
508midpoint sum
509upper sum
510lower sum
511partition
512subinterval
513norm of a partition
514limit of Riemann sums
515integrability
516trapezoidal rule
517Simpson's rule
518numerical integration
519fundamental theorem of calculus
520first fundamental theorem
521second fundamental theorem
522accumulation function
523net change theorem
524substitution
525u-substitution
526integration by parts
527partial fractions
528trigonometric substitution
529reduction formula
530improper integral
531convergent integral
532divergent integral
533area between two curves
534volume of revolution
535disc method
536washer method
537shell method
538cross-section method
539arc length
540surface of revolution
541average value of a function
542centroid of a region
543moment
544work
545fluid pressure
546consumer surplus
547separable differential equation
548slope field
549direction field
550initial condition
551particular solution
552Euler's method
553exponential growth model
554logistic growth model
12
Off the page: vectors and more dimensions
74 concepts · 72 drawable
By the endKeep the method when the plane runs out of room.
555three-dimensional space
556z-axis
557octant
558coordinate space
559right-handed system
560vector
561scalar
562magnitude
563direction
564component
565position vector
566unit vector
567zero vector
568vector addition
569scalar multiplication
570dot product
571scalar product
572cross product
573vector product
574projection
575angle between vectors
576orthogonality
577linear combination
578span
579basis
580linear independence
581matrix
582matrix transformation
583eigenvector
584eigenvalue
585line in space
586vector equation of a line
587parametric equations in space
588plane in space
589normal vector
590equation of a plane
591distance in three dimensions
592sphere
593cylinder
594cone
595quadric surface
596surface
597level curve
598contour
599contour map
600function of two variables
601partial derivative
602gradient vector
603directional derivative
604tangent plane
605critical point in 3D
606saddle point
607Lagrange multiplier
608double integral
609iterated integral
610region of integration
611order of integration
612Jacobian
613change of variables
614polar substitution
615cylindrical coordinates
616spherical coordinates
617triple integral
618vector field
619divergence
620curl
621line integral
622surface integral
623flux
624conservative field
625potential function
626Green's theorem
627Stokes' theorem
628divergence theorem
13
Where the picture stops working
53 concepts · 28 drawable
By the endSay when to stop trusting your eyes, and reach for a proof instead.