y -5 0 5 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 3 x = 3 x = 3: true at one input, false everywhere else
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 x² + 4 = 0 x² + 4 = 0: the empty solution set
y -30 -20 -10 0 10 20 n = 4 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 3x − 2 = 4 3x − 2 = 4: one crossing
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 2 x² − 4 = 0 x² − 4 = 0: two crossings
y -400 -200 0 200 400 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.73 0 1.73 x³ − 3x = 0 x³ − 3x = 0: three crossings
3/(x - 1) + 2/(x + 4) = 1: x = 1 and x = -4 excluded, solutions 1 plus or minus sqrt 15 x y −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 −4 −3 −2 −1 1 2 3 4 5 6 3/(x − 1) + 2/(x + 4) y = 1 x = 1 − √15 x = 1 + √15 x = −4 excluded x = 1 excluded Rule out x = 1 and x = −4 first; y = 1 is met at x = 1 ± √15 Excluded values first, solutions second
y 0 2 4 6 8 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -3 3 |x| = 3 |x| = 3: the V crosses the level twice
y 2 4 6 8 10 12 14 n = 2 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 |x| + 5 = 2 |x| + 5 = 2: no solution
y 0 2 4 6 8 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -3 3 |x| = 3 |x| = 3 has solutions x = −3 and x = 3
|x^2 - 4| = 5 at x = 3 and x = -3 only x y −4 −3 −2 −1 1 2 3 4 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 y = −5: never reached y = 5 x = −3 x = 3 dashed: x² − 4 |x² − 4| = 5 only at x = ±3: the case x² − 4 = −5 has no solution Two cases, one of them empty
|2x - 1| = |x + 4| at x = -1 and x = 5 x y −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 −1 1 2 3 4 5 6 7 8 9 10 11 12 |2x − 1| |x + 4| x = −1 x = 5 Equal magnitudes twice: 2x − 1 = x + 4 and 2x − 1 = −(x + 4) Two V-shapes, two crossings
|x - 2| = 3x - 4 holds only at x = 3/2; x = 1 is extraneous x y −1 1 2 3 4 −3 −2 −1 1 2 3 4 5 6 x = 1: extraneous |x − 2| 3x − 4 x = 3/2 |x − 2| = 1 3x − 4 = −1 Squaring keeps x = 1, but there |x − 2| = 1 while 3x − 4 = −1 Where squaring goes wrong
y 0 5 10 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 5 −x + 5 = 0 −x + 5 = 0: the line crosses the axis at x = 5
y 0 2 4 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 = 3 2 = 3: a horizontal line that never reaches the level
y -400 -200 0 200 400 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.73 0 1.73 x³ − 3x = 0 x³ − 3x = 0: degree 3, three real roots
The four roots of z^4 + 1 = 0: two conjugate pairs Re Im 1 −1 i −i (1 + i)/√2 45° (−1 + i)/√2 135° (−1 − i)/√2 225° (1 − i)/√2 315° a root and its conjugate: mirror images across the real axis z⁴ + 1 = 0 has no real root, but exactly 4 complex roots — one per degree, 90° apart on the unit circle, in two conjugate pairs
Degree 4, four roots, two conjugate pairs
2x^3 - 3x^2 - 8x + 12: of 16 rational candidates, only -2, 3/2 and 2 are roots x y −3 −2 −1 1 2 3 −6 −4 −2 2 4 6 8 10 12 14 16 18 candidates ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 roots: −2, 3/2, 2 P(x) = 2x³ − 3x² − 8x + 12: three of sixteen candidates are roots Candidates versus roots
Three quartics with four, two and no real roots x y −2 −1 1 2 −5 −4 −3 −2 −1 1 2 3 4 5 6 x⁴ − 5x² + 4: four roots x⁴ − 2x² − 3: two roots x⁴ + 1: none A quartic can have four, two or no real roots Four, two or no real roots
y -600 -400 -200 0 200 400 600 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0 x³ = 0 x³ = 0: one root of multiplicity three
y 0 20 40 60 80 n = 5 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 4 x² − 2x − 3 = 5 x² − 2x − 3 = 5: a quadratic not yet in standard form
(x - 2)(x - 3) = 0 exactly where x - 2 = 0 or x - 3 = 0 x x = 2: x − 2 = 0 x = 3: x − 3 = 0 x − 2 x − 3 (x − 2)(x − 3) = x² − 5x + 6 The product is zero exactly where one factor is zero Factoring turns one quadratic into two linear equations
x² 3x 3x 5 4 x +3 x +3 (x + 3)² − 4 = 0
x² + 6x + 5 = 0 becomes (x + 3)² − 4 = 0
2x^2 - 3x - 5 = 0: roots 3/4 plus or minus 7/4 x y −2 −1 1 2 3 4 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 axis x = −b/2a = 3/4 y = 2x² − 3x − 5 3/4 − 7/4 = −1 3/4 + 7/4 = 5/2 Roots = −b/2a ± √Δ/2a: 7/4 either side of the axis What the ± in the formula does
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 x² + 4 = 0 x² + 4 = 0: the parabola never reaches the axis
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 2 x² − 4 = 0 x² − 4 = 0: the solutions are the x-intercepts
x/(x - 3) and 9/(x - 3) + 2: x = 3 excluded, they meet at x = -3 x y −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 x = 3 excluded x = −3: both sides 1/2 x/(x − 3) 9/(x − 3) + 2 x = 3 is ruled out first; the two sides meet only at x = −3 The excluded value is a wall, not a solution
x/(x - 3) - 3/(x - 3) equals 1 for every x except 3 x y −2 −1 1 2 3 4 5 6 7 8 −1 1 2 3 x/(x − 3) − 3/(x − 3) x = 3: undefined = 1, the right side Clearing gives 0 = 0, yet the solution set is every x except 3 An identity with one point missing
2/(x^2 - 1) and 1/(x - 1) never meet; the candidate x = 1 is excluded x y −4 −3 −2 −1 1 2 3 4 −5 −4 −3 −2 −1 1 2 3 4 5 2/(x² − 1) 1/(x − 1) candidate x = 1 excluded x = −1 excluded The graphs never meet: the only candidate, x = 1, is excluded A candidate that lands on an excluded value
x² 3x 3x 5 gap = 4 x 3 x 3
x² + 6x + 5, frozen at the gap
x² 2x 2x 1.5 gap = 2.5 x 2 x 2
2x² + 8x + 3, frozen at the gap
x² -2x -2x 1 gap = 3 x -2 x -2
x² − 4x + 1, frozen at the gap
x² 2.5x 2.5x 2 gap = 4.25 x 2.5 x 2.5
x² + 5x + 2, frozen at the gap
x² 2x 2x 2.3333 gap = 1.6667 x 2 x 2
3x² + 12x + 7, frozen at the gap
x² 6x 5
Step 1: starting equation, frozen
Working with the monic part: x² + 4x (factor of 2 restored at the end)
Factor-out step (2x² + 8x + 3), frozen
x² x x
Step 2: the x² square, frozen
x² 3x 3x x 3 x 3
Step 3: splitting 6x, frozen
x² 3x 3x 5 corner needs 3 × 3 = 9 x 3 x 3
Step 4: the constant in the corner, frozen
x² 3x 3x 5 gap = 4 x 3 x 3
Step 5: the gap, frozen
x² 3x 3x 5 4 x +3 x +3 (x + 3)² − 4 = 0
Step 6: vertex form, frozen
x² 3x 3x 5 4 x +3 x +3 (x + 3)² − 4 = 0 → solve for x
Step 7: solving, frozen
a² b² ab ab a + b a + b a b a b
(a + b)² as an area: a² + ab + ab + b²
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 C(5, 2) = 10 — 10 paths from the top C(5, 2) = 10 paths through Pascal’s triangle
a² ab ac ab b² bc ac bc c² a a b b c c a+b+c a+b+c
(a + b + c)² as nine pieces
a · (a − b) (a − b) · b a + b a − b a b
a² − b² rearranged into (a + b)(a − b)
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) < 0 (x + 2)(x − 1)(x − 5) < 0: two stretches of the line
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) ≤ 0 The same expression with ≤: the boundaries join the set
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) < 0 (x + 2)(x − 1)(x − 5) < 0: the sign chart
-3x >= 12 holds exactly for x <= -4 x y −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 −8 −4 4 8 12 16 20 24 x = −4 y = −3x y = 12 line on or above 12: x ≤ −4 Dividing by −3 flips ≥ to ≤: −3x ≥ 12 holds only left of −4 Why dividing by a negative flips the sign
f(x) -10 0 10 x -4 -3 -2 -1 0 1 2 3 4 5 -2 3 (x + 2)(x − 3) < 0 (x + 2)(x − 3) < 0: between the roots
f(x) -100 0 100 x -5 -4 -3 -2 -1 0 1 2 3 4 -3 2 (x + 3)(x − 2)² < 0 (x + 3)(x − 2)² < 0: a double root does not change the sign
f(x) -5 0 5 x -4 -3 -2 -1 0 1 2 3 1 -2 (x − 1) / (x + 2) < 0 (x − 1)/(x + 2) < 0: two kinds of critical point
f(x) -2 0 2 x -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 3 |x| − 3 < 0 |x| − 3 < 0: the V dips below the axis on (−3, 3)
|x| < 3 is one interval; |x| > 3 is two rays |x| < 3: within 3 of 0 (−3, 3) −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 |x| > 3: more than 3 from 0 (−∞, −3) ∪ (3, ∞) included excluded Less-than traps x between two bounds; greater-than pushes it out One interval or two rays
f(x) -4 -2 0 2 4 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 6 |x − 2| − 4 < 0 |x − 2| − 4 < 0: the interval (−2, 6)
|x^2 - 7x + 10| >= 2 holds on three pieces x y 1 2 3 4 5 6 7 −1 1 2 3 4 5 6 7 8 y = 2 x ≤ 1.44, 3 ≤ x ≤ 4, x ≥ 5.56 [3, 4]: the fold tops 2 |x² − 7x + 10| ≥ 2 holds on three separate pieces Three pieces from one inequality
|x - 5| < 3 and |x - 1| < |x - 7| read as distances |x − 5| < 3: within 3 of 5 (2, 8) −2 −1 0 1 2 3 4 5 6 7 8 9 10 |x − 1| < |x − 7|: closer to 1 than to 7 x < 4 included excluded Distance from a point, or nearer of two points: 4 is the midpoint Absolute value as distance on the line
4x - 7 <= 2x + 11: the lines meet at x = 9 and the solution is x <= 9 x y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 −10 10 20 30 40 50 x = 9 4x − 7 2x + 11 left of 9: 4x − 7 is below 2x + 11 solution x ≤ 9, that is (−∞, 9] The boundary solves the equation; the inequality picks a side Where the boundary comes from
AND gives (-2, 3]; OR gives two rays −3 < 2x + 1 ≤ 7 (AND) (−2, 3] −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 x + 3 < −1 or x + 3 > 5 (OR) (−∞, −4) ∪ (2, ∞) included excluded AND keeps the overlap, one interval; OR keeps either, two rays AND versus OR on the number line
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) < 0 Three simple roots, four intervals, alternating signs
f(x) -100 0 100 x -5 -4 -3 -2 -1 0 1 2 3 4 -3 2 (x + 3)(x − 2)² < 0 The double root at x = 2 keeps the sign
x^3 - 2 > 0 exactly for x greater than the cube root of 2 x y −2 −1 1 2 3 −10 −5 5 10 15 y = x³ − 2 ∛2 ≈ 1.26 x³ − 2 > 0 only right of ∛2 solution (∛2, ∞) An irrational root is still a boundary: x³ − 2 > 0 for x > ∛2 A boundary that is not a rational number
(x + 2)^2 (x - 1)(x - 4) > 0: touches at -2, crosses at 1 and 4 x y −3 −2 −1 1 2 3 4 5 −60 −40 −20 20 40 (x + 2)²(x − 1)(x − 4) −2: double root, the sign does not change 1 and 4: simple roots, the sign flips solution: x < −2, −2 < x < 1, x > 4 Positive at both ends; the double root −2 keeps the sign Reading the signs from the ends inward
x^2 < 3x + 10: the parabola is below the line for -2 < x < 5 x y −4 −3 −2 −1 1 2 3 4 5 6 7 −4 4 8 12 16 20 24 28 32 36 y = x² y = 3x + 10 x² below the line: −2 < x < 5 x² < 3x + 10, or x² − 3x − 10 < 0: solution −2 < x < 5 Before and after standard form
f(x) -10 0 10 x -4 -3 -2 -1 0 1 2 3 4 5 -2 3 (x + 2)(x − 3) < 0 Δ > 0: the parabola dips between its two roots
(x - 2)^2 > 0 for every x except 2 x y −1 1 2 3 4 5 −2 −1 1 2 3 4 5 6 7 8 9 (x − 2)² > 0: every x except 2 ≥ 0: every real x Δ = 0: touches the axis once, positive everywhere else A single root that never changes the sign
f(x) -20 -10 0 10 20 x -5 -4 -3 -2 -1 0 1 2 3 4 5 (x² + 4) < 0 Δ < 0: x² + 4 never reaches the axis
x^4 - 5x^2 + 4 > 0 on three pieces x y −2 −1 1 2 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 u = x²: (u − 1)(u − 4) > 0, so u < 1 or u > 4 solution x < −2, −1 < x < 1, x > 2 x⁴ − 5x² + 4 > 0 on three pieces, found through u = x² A quartic inequality solved as a quadratic
f(x) -5 0 5 x -4 -3 -2 -1 0 1 2 3 1 -2 (x − 1) / (x + 2) < 0 Numerator zero at 1, pole at −2
(x - 3)/(x + 2) > 1 holds exactly for x < -2 x y −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 (x − 3)/(x + 2) y = 1 above 1 for x < −2 multiplying by x + 2 misses it (x − 3)/(x + 2) > 1 holds exactly for x < −2 What cross-multiplying gets wrong
(7 - x)/((x - 3)(x + 1)) <= 0 on (-1, 3) and [7, inf) x y −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 11 12 −3 −2 −1 1 2 3 −1, 3: excluded 7: included solution: −1 < x < 3 or x ≥ 7 Denominator zeros are walls; the numerator zero 7 joins ≤ 0 Two kinds of critical points on one graph
Base 2, max power 10 — the default load Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 Each row is ×2 the row above it — that is the power of exponents! Powers of 2: read the table backwards for log₂
log base 2, e and 10 all pass through (1, 0) and through (a, 1) x y 1 2 3 4 5 6 7 8 9 10 11 −3 −2 −1 1 2 3 4 log₂ x ln x log₁₀ x each passes through (a, 1) and every one passes through (1, 0) log_a(1) = 0 and log_a(a) = 1 for every base Two points every logarithm shares
2^x and log base 2 of x are reflections in y = x x y −2 −1 1 2 3 4 5 6 7 −1 1 2 3 4 5 y = 2ˣ y = log₂ x (2, 4) (4, 2) y = x log₂(2ˣ) = x: (2, 4) on 2ˣ becomes (4, 2) on log₂ x The logarithm undoes the exponential
ln x and log x: both through (1, 0); ln x is log x stretched by ln 10 x y 1 2 3 4 5 6 7 8 9 10 11 12 −2 −1 1 2 3 ln x log x (e, 1) (10, 1) (10, ln 10 ≈ 2.303) at every x: ln x ≈ 2.303 · log x Same shape: ln x is log x stretched by ln 10 ≈ 2.303 One curve, two vertical scales
log2 x + log2(x - 2) = 3 only at x = 4; the curve exists for x > 2 x y −3 −2 −1 1 2 3 4 5 6 7 8 9 −3 −2 −1 1 2 3 4 5 6 log₂ x + log₂(x − 2) y = 3 x = 4 domain: x > 2 x = −2 is rejected: log₂(−2) is undefined Only x = 4 survives: the left side exists only for x > 2 Why x = −2 is thrown out
(log2 x)^2 - 5 log2 x + 6 = 0 at x = 4 and x = 8 x y 1 2 3 4 5 6 7 8 9 10 −1 1 2 3 4 5 6 7 8 (log₂ x)² − 5 log₂ x + 6 u = log₂ x: u² − 5u + 6 = 0, so u = 2 or 3 x = 2² = 4 or x = 2³ = 8 Quadratic in log₂ x: zeros at x = 4 and x = 8 A quadratic hiding inside a logarithm
2^(x + 3) = 5^(x - 1) at x approximately 4.026 x y 1 2 3 4 5 50 100 150 200 250 300 2ˣ⁺³ 5ˣ⁻¹ x ≈ 4.026 x = (ln 5 + 3 ln 2)/(ln 5 − ln 2) ≈ 4.026 2ˣ⁺³ = 5ˣ⁻¹: the faster-growing 5ˣ⁻¹ overtakes at x ≈ 4.026 Two exponentials with different bases
log2 x: each doubling of x adds 1 to the output x y 1 2 3 4 5 6 7 8 9 −3 −2 −1 1 2 3 4 y = log₂ x (1, 0) (2, 1) (4, 2) (8, 3) each doubling of x adds 1 to y log₂ x rises without bound, but ever more slowly Equal steps from doubling
log2 x falls to minus infinity and log1/2 x rises to plus infinity as x approaches 0 x y −1 1 2 3 4 5 6 −4 −3 −2 −1 1 2 3 4 log₂ x log₁/₂ x log₂ x → −∞ as x → 0⁺ log₁/₂ x → +∞ as x → 0⁺ The y-axis is a vertical asymptote: the graphs never reach x = 0 Opposite directions at the same wall
log2(x - 3): log2 x moved 3 right, asymptote x = 3 x y −1 1 2 3 4 5 6 7 8 9 10 −3 −2 −1 1 2 3 4 x = 3 y = log₂(x − 3) (4, 0) (5, 1) dashed: y = log₂ x Shifting right 3 moves the asymptote to x = 3 and (1, 0) to (4, 0) Moving the asymptote
log3 x + 2: log3 x moved up 2, asymptote unchanged x y 1 2 3 4 5 6 7 8 9 10 −3 −2 −1 1 2 3 4 5 y = log₃ x y = log₃ x + 2 (1, 2) (3, 3) every point moves up 2; the asymptote stays at x = 0 A vertical shift moves the anchors, not the asymptote Lifting the curve
2 log2 x: log2 x stretched vertically by 2, (1, 0) fixed x y 1 2 3 4 5 6 7 8 9 −4 −3 −2 −1 1 2 3 4 5 6 7 y = log₂ x y = 2 log₂ x (1, 0) stays fixed (2, 1) (2, 2) Every height doubles; only the zero at x = 1 cannot move A vertical stretch by 2
-log2 x reflects across the x-axis; log2(-x) reflects across the y-axis x y −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 −3 −2 −1 1 2 3 dashed: y = log₂ x y = −log₂ x y = log₂(−x) (2, −1) (−2, 1) −log₂ x flips across the x-axis; log₂(−x) flips across the y-axis Two different reflections
log4(x + 2) < 0 on -2 < x < -1 x y −3 −2 −1 1 2 3 −3 −2 −1 1 2 x = −2: domain edge y = log₄(x + 2) x = −1 below 0 for −2 < x < −1 log₄(x + 2) < 0 only on −2 < x < −1 The domain closes the interval
1 < log2 x < 4 exactly for 2 < x < 16 x y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 −2 −1 1 2 3 4 5 y = 1 y = 4 y = log₂ x (2, 1) (16, 4) solution 2 < x < 16 Squeezing log₂ x between 1 and 4 squeezes x between 2¹ and 2⁴ Bounding a logarithm on both sides
log2 x > 1 for x > 2, but log1/2 x > 1 for 0 < x < 1/2 x y 1 2 3 4 5 −3 −2 −1 1 2 3 log₂ x log₁/₂ x y = 1 x = 2 x = 1/2 log₂ x > 1 for x > 2 log₁/₂ x > 1 for 0 < x < 1/2 Same inequality, opposite sides: the base decides the direction Why the base decides the direction
log2(x - 3) is defined only for x > 3; log5(x^2 + 1) for every x x y −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −3 −2 −1 1 2 3 4 x = 3 log₂(x − 3): x > 3 only log₅(x² + 1): every x x − 3 > 0 needs x > 3; x² + 1 is always positive The argument sets the domain
Horizontal line test: log2 x passes, log2(x^2 + 1) fails log₂ x x y y = 1.5 y = −0.5 y = −2.5 one crossing per line: one-to-one log₂(x² + 1) x y y = 2 y = 1 x = ±1 both give 1: not one-to-one Always increasing means one-to-one; log₂(x² + 1) turns back at 0 One-to-one, seen with horizontal lines
log2 32 = log2 8 + log2 4: multiplying inputs adds outputs x y 4 8 12 16 20 24 28 32 −1 1 2 3 4 5 6 y = log₂ x log₂ 4 = 2 log₂ 8 = 3 log₂ 32 = 5 = 3 + 2 8 × 4 = 32, and 3 + 2 = 5 Multiplying inputs adds outputs: log₂(8 · 4) = log₂ 8 + log₂ 4 The product rule on the graph
(x + 2)(x + 3), all four cells delivered x 3 x x² 3x 2 2x 6 LIKE-TERM BUCKETS x² terms 1 sum: x² x terms 3 + 2 sum: 5x constants 6 sum: 6 P(x) · Q(x) = x² + 5x + 6
(x + 2)(x + 3) = x² + 5x + 6 on the grid
x² 3x 3x 5 corner needs 3 × 3 = 9 x 3 x 3
x² + 6x + 9: the corner that completes the square
f(x) -100 0 100 x -5 -4 -3 -2 -1 0 1 2 3 4 -3 2 (x + 3)(x − 2)² < 0 A simple root crosses, a double root touches
y 0 20 40 60 80 n = 5 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 4 x² − 2x − 3 = 5 x² − 2x − 3 against the level y = 5
y -400 -200 0 200 400 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.30 -1 2.30 x³ − 4x = 3 x³ − 4x = 3: opposite ends, two turning points
(x² − 3x + 2)(2x + 5), all six cells delivered 2x 5 x² 2x³ 5x² −3x −6x² −15x 2 4x 10 LIKE-TERM BUCKETS x³ terms 2 sum: 2x³ x² terms 5 − 6 sum: −x² x terms − 15 + 4 sum: −11x constants 10 sum: 10 P(x) · Q(x) = 2x³ − x² − 11x + 10
(x² − 3x + 2)(2x + 5): six cells, four buckets
(x + 1)(x² − x + 1), all six cells delivered x² −x 1 x x³ −x² x 1 x² −x 1 LIKE-TERM BUCKETS x³ terms 1 sum: x³ x² terms − 1 + 1 sum: 0 x terms 1 − 1 sum: 0 constants 1 sum: 1 P(x) · Q(x) = x³ + 1
(x + 1)(x² − x + 1): the middle buckets cancel
(x + 2)(x + 3), all four cells delivered x 3 x x² 3x 2 2x 6 LIKE-TERM BUCKETS x² terms 1 sum: x² x terms 3 + 2 sum: 5x constants 6 sum: 6 P(x) · Q(x) = x² + 5x + 6
First, Outer, Inner, Last: the four cells of (x + 2)(x + 3)
y -400 -200 0 200 400 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.73 0 1.73 x³ − 3x = 0 x³ − 3x = x(x − √3)(x + √3): three roots, three crossings
y -600 -400 -200 0 200 400 600 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0 x³ = 0 x³ = 0: the root 0 appears three times
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 2 x² − 4 = 0 x² − 4 = 0: the real roots are the x-intercepts
(x + 2)(x + 3), all four cells delivered x 3 x x² 3x 2 2x 6 LIKE-TERM BUCKETS x² terms 1 sum: x² x terms 3 + 2 sum: 5x constants 6 sum: 6 P(x) · Q(x) = x² + 5x + 6
(x + 2)(x + 3), frozen at completion
(2x − 1)(x + 4), all four cells delivered x 4 2x 2x² 8x −1 −x −4 LIKE-TERM BUCKETS x² terms 2 sum: 2x² x terms 8 − 1 sum: 7x constants − 4 sum: −4 P(x) · Q(x) = 2x² + 7x − 4
(2x − 1)(x + 4), frozen at completion
(x² − 3x + 2)(2x + 5), all six cells delivered 2x 5 x² 2x³ 5x² −3x −6x² −15x 2 4x 10 LIKE-TERM BUCKETS x³ terms 2 sum: 2x³ x² terms 5 − 6 sum: −x² x terms − 15 + 4 sum: −11x constants 10 sum: 10 P(x) · Q(x) = 2x³ − x² − 11x + 10
(x² − 3x + 2)(2x + 5), frozen at completion
(x + 1)(x² − x + 1), all six cells delivered x² −x 1 x x³ −x² x 1 x² −x 1 LIKE-TERM BUCKETS x³ terms 1 sum: x³ x² terms − 1 + 1 sum: 0 x terms 1 − 1 sum: 0 constants 1 sum: 1 P(x) · Q(x) = x³ + 1
(x + 1)(x² − x + 1), frozen at completion
(2x² + x − 3)(x² − 2x + 1), all nine cells delivered x² −2x 1 2x² 2x⁴ −4x³ 2x² x x³ −2x² x −3 −3x² 6x −3 LIKE-TERM BUCKETS x⁴ terms 2 sum: 2x⁴ x³ terms − 4 + 1 sum: −3x³ x² terms 2 − 2 − 3 sum: −3x² x terms 1 + 6 sum: 7x constants − 3 sum: −3 P(x) · Q(x) = 2x⁴ − 3x³ − 3x² + 7x − 3
(2x² + x − 3)(x² − 2x + 1), frozen at completion
Base 2, max power 10 — the default load Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 Each row is ×2 the row above it — that is the power of exponents! Powers of 2 from 2⁰ to 2¹⁰
Base 2, max power 10 — the default load Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 Each row is ×2 the row above it — that is the power of exponents! 2³ · 2⁴ = 2⁷: add the rows
Base 2, max power 10 — the default load Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 Each row is ×2 the row above it — that is the power of exponents! aⁿ as a table: exponent, expanded product, value
Base 5, max power 10 — the first base capped at 10 Power Expression Value 5⁰ 1 1 5¹ 5 5 5² 5 × 5 25 5³ 5 × 5 × 5 125 5⁴ 5 × 5 × 5 × 5 625 5⁵ 5 × 5 × 5 × 5 × 5 3,125 5⁶ 5 × 5 × 5 × 5 × 5 × 5 15,625 5⁷ 5 × 5 × 5 × 5 × 5 × 5 × 5 78,125 5⁸ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 390,625 5⁹ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 1,953,125 5¹⁰ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 9,765,625 Each row is ×5 the row above it — that is the power of exponents! Powers of 5: the table starts at 5⁰ = 1
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 C(5, 2) = 10 — 10 paths from the top C(5, 2) = 10 is the fourth triangular number
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 C(5, 2) = 10 — 10 paths from the top T₄ = C(5, 2) = 10
a b a b a b a b a b a b a b start aaa aab aba abb baa bab bba bbb a³ aaa paths landing here: 1 a²b aab, aba, baa paths landing here: 3 ab² abb, bab, bba paths landing here: 3 b³ bbb paths landing here: 1
Decision Tree, n = 3, frozen
a + b · a + b · a + b b · b · b = b³ (8 / 8 delivered) a³ aaa count: 1 a²b aab aba baa count: 3 ab² abb bab bba count: 3 b³ bbb count: 1 (a + b)³ = a³ + 3a²b + 3ab² + b³
Distribution, n = 3, complete
1 1 1 1 2 1 1 3 3 1 C(3, 1) = 3 — 3 paths from the top Pascal Paths, cell (3, 1) engaged
a b start a b a a paths landing here: 1 b b paths landing here: 1
Decision Tree, n = 1, frozen
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 C(5, 2) = 10 — 10 paths from the top Pascal Paths, cell (5, 2) engaged
(a+b)² a + b a + b
Step 1: the (a+b) square, frozen
a + b a + b a b a b
Step 2: the side split, frozen
a + b a + b a b a b
Step 3: the grid cuts, frozen
a² b² ab ab a + b a + b a b a b
Step 4: the four pieces, frozen
a² a a
Step 1: the a² square, frozen
a² b² b a − b a a − b b a
Step 2: the b² corner, frozen
(a−b)² ab b² ab b a − b a a − b b a
Step 3: the overlapping strips, frozen
(a−b)² ab b² ab b a − b a a − b b a
Step 4: the discard, frozen at full separation
(a+b+c)² a+b+c a+b+c
Step 1: the (a+b+c) square, frozen
a a b b c c a+b+c a+b+c
Step 2: the three-way split, frozen
a² ab ac ab b² bc ac bc c² a a b b c c a+b+c a+b+c
Step 3: the 3×3 grid, frozen
a² ab ac ab b² bc ac bc c² a a b b c c
Step 4: the explosion, frozen at full separation
a² a a
Step 1: the a² square, frozen
b² a² a a − b b b a − b
Step 2: the removal mark, frozen
b² (removed) (a − b) · b a · (a − b) a a − b b b a − b
Step 3: the L-shape, frozen
a · (a − b) (a − b) · b a + b a − b a b
Step 4: the rectangle, frozen settled
y -5 0 5 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 3 x = 3 x = 3, frozen
y -30 -20 -10 0 10 20 n = 4 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 3x − 2 = 4 3x − 2 = 4, frozen
y 0 5 10 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 5 −x + 5 = 0 −x + 5 = 0, frozen
y 0 2 4 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 = 3 2 = 3 (a = 0), frozen
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 2 x² − 4 = 0 x² − 4 = 0, frozen
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0 x² = 0 x² = 0, frozen
y 0 20 40 60 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 x² + 4 = 0 x² + 4 = 0, frozen
y 0 20 40 60 80 n = 5 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 4 x² − 2x − 3 = 5 x² − 2x − 3 = 5, frozen
y -400 -200 0 200 400 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.73 0 1.73 x³ − 3x = 0 x³ − 3x = 0, frozen
y -600 -400 -200 0 200 400 600 n = 0 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0 x³ = 0 x³ = 0, frozen
y -400 -200 0 200 400 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -1.30 -1 2.30 x³ − 4x = 3 x³ − 4x = 3, frozen
y 0 2 4 6 8 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -3 3 |x| = 3 |x| = 3, frozen
y 2 4 6 8 10 12 n = 3 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 0 |x| + 3 = 3 |x| + 3 = 3, frozen
y 2 4 6 8 10 12 14 n = 2 x -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 |x| + 5 = 2 |x| + 5 = 2, frozen
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) < 0 Operator <, frozen
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) ≤ 0 Operator ≤, frozen
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) > 0 Operator >, frozen
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) ≥ 0 Operator ≥, frozen
f(x) -100 0 100 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 -2 1 5 (x + 2)(x − 1)(x − 5) < 0 (x+2)(x−1)(x−5) < 0, frozen
f(x) -100 0 100 x -5 -4 -3 -2 -1 0 1 2 3 4 -3 2 (x + 3)(x − 2)² < 0 (x+3)(x−2)² < 0, frozen
f(x) -20 0 20 x -3 -2 -1 0 1 2 3 -1 0 1 (x + 1)(x)(x − 1) < 0 (x+1)(x)(x−1) < 0, frozen
f(x) -10 0 10 x -4 -3 -2 -1 0 1 2 3 4 5 -2 3 (x + 2)(x − 3) < 0 x² − x − 6 < 0, frozen
f(x) -20 -10 0 10 20 x -5 -4 -3 -2 -1 0 1 2 3 4 5 (x² + 4) < 0 x² + 4 < 0, frozen
f(x) -10 0 10 x -3 -2 -1 0 1 2 3 4 5 -1 3 (x + 1)(x − 3) < 0 −x² + 2x + 3 < 0, frozen
f(x) -2 0 2 x -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 3 |x| − 3 < 0 |x| − 3 < 0, frozen
f(x) -4 -2 0 2 4 x -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 6 |x − 2| − 4 < 0 |x − 2| − 4 < 0, frozen
f(x) -2 -1 0 1 2 x -3 -2 -1 0 1 -1 |x + 1| < 0 |x + 1| < 0, frozen
f(x) -5 0 5 x -4 -3 -2 -1 0 1 2 3 1 -2 (x − 1) / (x + 2) < 0 (x−1)/(x+2) < 0, frozen
f(x) -10 -5 0 5 10 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -3 4 (x + 3) / (x − 4) < 0 (x+3)/(x−4) < 0, frozen
f(x) -4 -2 0 2 4 x 0 1 2 3 4 5 2 3 (x − 2) / (x − 3) < 0 (x−2)/(x−3) < 0, frozen
f(x) -1 0 1 x -2 -1 0 1 2 3 4 5 6 0 4 √x − 2 < 0 √x − 2 < 0, frozen
f(x) -1 -0.5 0 0.5 1 x -5 -4 -3 -2 -1 0 -3 -2 √(x + 3) − 1 < 0 √(x+3) − 1 < 0, frozen
f(x) -2 0 2 x -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 17 √(x − 1) − 4 < 0 √(x−1) − 4 < 0, frozen
Base 2, max power 10 — the default load Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 Each row is ×2 the row above it — that is the power of exponents! Base 2, max power 10, frozen
Base 10, max power 10 — place value, one zero per row Power Expression Value 10⁰ 1 1 10¹ 10 10 10² 10 × 10 100 10³ 10 × 10 × 10 1,000 10⁴ 10 × 10 × 10 × 10 10,000 10⁵ 10 × 10 × 10 × 10 × 10 100,000 10⁶ 10 × 10 × 10 × 10 × 10 × 10 1,000,000 10⁷ 10 × 10 × 10 × 10 × 10 × 10 × 10 10,000,000 10⁸ 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 100,000,000 10⁹ 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 1,000,000,000 10¹⁰ 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 10,000,000,000 Each row is ×10 the row above it — that is the power of exponents! Base 10, max power 10, frozen
Base 5, max power 10 — the first base capped at 10 Power Expression Value 5⁰ 1 1 5¹ 5 5 5² 5 × 5 25 5³ 5 × 5 × 5 125 5⁴ 5 × 5 × 5 × 5 625 5⁵ 5 × 5 × 5 × 5 × 5 3,125 5⁶ 5 × 5 × 5 × 5 × 5 × 5 15,625 5⁷ 5 × 5 × 5 × 5 × 5 × 5 × 5 78,125 5⁸ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 390,625 5⁹ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 1,953,125 5¹⁰ 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 9,765,625 Each row is ×5 the row above it — that is the power of exponents! Base 5, max power 10, frozen
Base 2, max power 16 — the small-base extended range Power Expression Value 2⁰ 1 1 2¹ 2 2 2² 2 × 2 4 2³ 2 × 2 × 2 8 2⁴ 2 × 2 × 2 × 2 16 2⁵ 2 × 2 × 2 × 2 × 2 32 2⁶ 2 × 2 × 2 × 2 × 2 × 2 64 2⁷ 2 × 2 × 2 × 2 × 2 × 2 × 2 128 2⁸ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 256 2⁹ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 2¹⁰ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 1,024 2¹¹ 2 × 2 × ... × 2 (11 times) 2,048 2¹² 2 × 2 × ... × 2 (12 times) 4,096 2¹³ 2 × 2 × ... × 2 (13 times) 8,192 2¹⁴ 2 × 2 × ... × 2 (14 times) 16,384 2¹⁵ 2 × 2 × ... × 2 (15 times) 32,768 2¹⁶ 2 × 2 × ... × 2 (16 times) 65,536 Each row is ×2 the row above it — that is the power of exponents! Base 2, max power 16, frozen
Base 7, max power 10 — last digits cycling 7, 9, 3, 1 Power Expression Value 7⁰ 1 1 7¹ 7 7 7² 7 × 7 49 7³ 7 × 7 × 7 343 7⁴ 7 × 7 × 7 × 7 2,401 7⁵ 7 × 7 × 7 × 7 × 7 16,807 7⁶ 7 × 7 × 7 × 7 × 7 × 7 117,649 7⁷ 7 × 7 × 7 × 7 × 7 × 7 × 7 823,543 7⁸ 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 5,764,801 7⁹ 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 40,353,607 7¹⁰ 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 282,475,249 Each row is ×7 the row above it — that is the power of exponents! Base 7, max power 10, frozen