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Prerequisite Facts and Formulas
AP Calculus AB
Line Equations (slope m)
Parabola Equations
Standard: y = ax2 + bx + c
Vertex (h,k): y = a(x – h)2 + k
Center: (h,k) Radius: r
x=
Quadratic Factoring with Squares
(x – h)2 + (y – k)2 = r2
−𝐛±√𝐛𝟐 −𝟒𝐚𝐜
𝟐𝐚
a2 – b2 = (a + b)(a – b)
a2 + 2ab + b2 = (a + b)2
a3 + b3 = (a + b)(a2 – ab + b2)
x0 = 1
a3 – b3 = (a – b)(a2 + ab + b2)
𝐱𝐦
xmxn = xm + n
𝐚 𝐱
(ab)x = axbx
𝐚𝐱
bx = a ⇒ logb(a) = x
Logarithm Product and Quotient
logb(mn) = logb(m) + logb(n)
Power Inside a Logarithm
𝐲
𝐦
𝐦⁄
𝐧
ln(x) = loge(x)
𝐦
logb( 𝐧 ) = logb(m) – logb(n)
𝐥𝐨𝐠 ( 𝐚)
𝐱
60°
x
45°
x√𝟐
2x
x
30°
45°
x
sin-1(
Tangent: 𝐱
𝐧
√𝐱 𝐦 = ( √𝐱 ) = 𝐱
logb(a) = 𝐥𝐨𝐠𝐱 (𝐛)
Right Triangle Trigonometry
Cosine: x-value
𝐧
logb(am) = mlogb(a)
sin(θ) =
Sine: y-value
(xm)n = xmn
log(x) = log10(x)
Change of Base (any non-zero x)
Unit Circle Coordinates (Quadrant I)
= xm – n
𝐱𝐧
(𝐛) = 𝐛 𝐱
Logarithms
Special Right Triangles
Based on the Pythagorean Theorem:
a2 + b2 = c2 (where a and b are leg
lengths and c is hypotenuse length)
𝟏
Point-Slope: y – y1 = m(x – x1)
Quadratic Formula
Exponent Rules (x ≠ 0)
𝟐
Slope-Intercept: y = mx + b
Standard Form Equation for Circles
Cubic Factoring
𝐲 −𝐲
m = 𝐱𝟐 −𝐱𝟏
from point (x1,y1) to (x2,y2)
Slope Formula
𝐚
𝐜
𝐚
𝐜
)=θ
x√𝟑
cos(θ) =
𝐛
𝐜
𝐛
cos-1(
Radians
0
x-value
1
y-value
0
𝐜
tan(θ) =
)=θ
𝝅
𝟔
√𝟑
𝟐
𝟏
𝟐
√𝟑
𝐚
𝐛
c
𝐚
tan-1( ) = θ
𝐛
𝝅
𝟒
√𝟐
𝟐
√𝟐
𝟐
a
θ
b
𝝅
𝟑
𝟏
𝟐
√𝟑
𝟐
𝝅
𝟐
0
1
Reciprocals for Sine and Cosine
csc x = 𝐬𝐢𝐧 𝐱
𝟏
sec x = 𝐜𝐨𝐬 𝐱
Tangent and Cotangent
tan x = 𝐜𝐨𝐬 𝐱
𝐬𝐢𝐧 𝐱
cot x = 𝐬𝐢𝐧 𝐱
Odd Trigonometric Functions
Even Trigonometric Functions
Pythagorean Trig Identities
Co-Function Trig Identities
csc(-x) = -csc(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
sec2x = 1 + tan2x
𝝅
sec x = csc(𝟐 – x)
𝝅
csc x = sec( 𝟐 – x)
sin x = cos( 𝟐 – x)
cos x = sin( 𝟐 – x)
tan x = cot(𝟐 – x)
𝝅
𝝅
cot x = tan( 𝟐 – x)
𝝅
y = sin-1x: y ∈ [− 𝟐 , 𝟐 ]
𝝅 𝝅
y = cos-1x: y ∈ [𝟎, 𝝅]
y = tan-1x: y ∈ (− 𝟐 , 𝟐 )
𝝅 𝝅
y = cot-1x: y ∈ (𝟎, 𝝅)
Axis: k
𝛑
𝛑 𝛑
y = csc-1x: y ∈ [− 𝟐 , 𝟐] , 𝐱 ≠ 𝟎
𝟐
Phase Displacement: h Amplitude: A
y = A cos B(x – h) + k
(base = b height = h)
Triangle Area
(base = b height = h)
Trapezoid Area
(bases: b1,b2, height = h)
Prism and Pyramid Volume
(area of base = B, height = h)
Cylinder and Cone Volume
(radius = r, height = h)
Cylinder Surface Area
(radius = r, height = h)
Cone Surface Area
(radius = r, slant-height = s)
Sphere Volume and Surface Area
(radius = r)
Scale Factor: Linear, Area, Volume
𝝅
cos(A ± B) = cos A cos B ∓ sin A sin B
Parallelogram Area
Distance Formula Between Points
csc2x = 1 + cot2x
sin(A ± B) = sin A cos B ± cos A sin B
y = sec-1x: y ∈ [𝟎, 𝛑], 𝐱 ≠
Sinusoid Function
𝐜𝐨𝐬 𝐱
sin(-x) = -sin(x)
sin2x + cos2x = 1
Sum/Difference for Sine and Cosine
Inverse Trig Principle Branches
𝟏
Period:
bh
𝟏
𝐛𝐡
∙bh or
𝟐
𝟐
𝟏
∙(b1 + b2)∙h
𝟐
𝟏
Pyramid: ∙Bh
𝟑
𝟏
Cylinder: πr2h Cone: ∙πr2h
𝟑
Prism: Bh
2πr2 + 2πrh
πr2 + πrs
V=
𝟒
𝟑
πr3
and
S.A. = 4πr2
d = √(𝐱 𝟐 − 𝐱 𝟏 )𝟐 + (𝐲𝟐 − 𝐲𝟏 )𝟐
L.S.F. = s
then
A.S.F = s2
and
V.S.F = s3
𝟐𝛑
𝐁