Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is picking a form that doesn't match the given data. In reality, slope-intercept y=mx+c, point-slope y−y₁=m(x−x₁), and two-point forms describe the same line.
Memory trick: use point-slope when you have a point and a slope.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope from two points. Use it when x₁ ≠ x₂.
m = (y₂−y₁)/(x₂−x₁)
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Distance from a point to a line. Use it when line ax+by+c=0.
d = |ax₀+by₀+c|/√(a²+b²)
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Parallel lines never meet and share the same slope; perpendicular lines cross at 90° with slopes multiplying to −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope-intercept (y=mx+c) is best when you know the slope and y-intercept; point-slope is best when you know the slope and any one point.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Equal slopes make lines parallel; perpendicular slopes multiply to −1.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A number e that classifies conics: circle e=0, ellipse e<1, parabola e=1, hyperbola e>1.
Memory trick: as e grows from 0, the shape opens up: circle → ellipse → parabola → hyperbola.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is mixing up which conic each eccentricity range gives. In reality, a number e that classifies conics: circle e=0, ellipse e<1, parabola e=1, hyperbola e>1.
Memory trick: as e grows from 0, the shape opens up: circle → ellipse → parabola → hyperbola.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A conic is the set of points whose distances to a focus and a directrix are in the fixed ratio e.
Memory trick: focus is a point inside; directrix is a line outside.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is confusing the focus (a point) with the directrix (a line). In reality, a conic is the set of points whose distances to a focus and a directrix are in the fixed ratio e.
Memory trick: focus is a point inside; directrix is a line outside.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
Circle (x−h)²+(y−k)²=r² and parabola y²=4ax are the building blocks.
Memory trick: y²=4ax opens right; x²=4ay opens up.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is mismatching the axis of the parabola with its equation. In reality, circle (x−h)²+(y−k)²=r² and parabola y²=4ax are the building blocks.
Memory trick: y²=4ax opens right; x²=4ay opens up.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
Equation of a circle with centre (h,k), radius r. Use it when any circle.
(x−h)²+(y−k)² = r²
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
Standard parabola opening rightward. Use it when vertex at the origin.
y² = 4ax
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
An ellipse is a closed curve with e<1 (sum of focal distances constant); a hyperbola is an open two-branch curve with e>1 (difference of focal distances constant).
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A parabola has eccentricity exactly 1 and one focus; a hyperbola has e>1 and two branches with two foci.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
Remember: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
A frequent error is assuming the limit always equals the function's value at that point. In reality, the value a function approaches as the input nears a point, even if it isn't defined there.
Memory trick: a limit can exist where f(a) doesn't — it's about approaching, not arriving.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
A frequent error is thinking every function is continuous everywhere. In reality, a function is continuous at a point if its limit there equals its actual value (no breaks or jumps).
Memory trick: continuous means you can draw it without lifting your pen.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
Lim(x→0) sin x / x = 1 is a key result used everywhere.
Memory trick: 0/0 is indeterminate — use standard limits or L'Hôpital's rule.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
A frequent error is evaluating it as 0/0 and stopping. In reality, lim(x→0) sin x / x = 1 is a key result used everywhere.
Memory trick: 0/0 is indeterminate — use standard limits or L'Hôpital's rule.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
A fundamental trigonometric limit. Use it when x measured in radians.
lim(x→0) sin x / x = 1
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
The limit is what f(x) approaches near a point; the value is what f actually equals there — they can differ at a discontinuity.
“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”
A continuous function has no breaks (limit = value everywhere); a discontinuous one has jumps, holes, or asymptotes.