SAT Math Formula Sheet & Desmos Power Guide

A complete SAT math reference with three worked examples for every formula—plus Desmos calculator strategies.
Click any flashcard to flip and reveal the formula and examples.

Algebra & Linear Equations

Slope
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Examples
(1,2) & (3,6): \( m = 2 \)
(-2,5) & (4,-1): \( m = -1 \)
(a,2a) & (a+3,5a): \( m = a \)
Slope-Intercept Form
\( y = mx + b \)
Examples
Slope 2, y-int 1: \( y=2x+1 \)
Through (0,-3), slope -4: \( y=-4x-3 \)
(2,5) & (6,13): \( y=2x+1 \)
Point-Slope Form
\( y - y_1 = m(x - x_1) \)
Examples
Slope 3, (1,2): \( y-2=3(x-1) \)
Slope -2, (4,-5): \( y+5=-2(x-4) \)
Slope m, (h,k): \( y-k=m(x-h) \)
Midpoint
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Examples
(2,4)&(4,8): (3,6)
(-3,7)&(5,-1): (1,3)
(a,b)&(c,d): \( \left( \frac{a+c}{2}, \frac{b+d}{2} \right) \)
Distance
\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
Examples
(0,0)-(3,4): 5
(1,2)-(4,6): 5
(a,b)-(a+3,b+4): 5
Parallel Lines
Equal slopes
Examples
y=2x+1 & y=2x-3
3x-y=7 & 6x-2y=4
All lines with m=5
Perpendicular Lines
Slopes are negative reciprocals (\( m_1 m_2 = -1 \))
Examples
m=2 & m=-1/2
y=3x+1 & y=-1/3x+2
y=mx+b & y=-1/m x+c
Standard Form
\( Ax + By = C \)
Examples
2x+3y=6
5x-y=10
4x+0y=8 (vertical line)

Quadratics & Polynomials

Quadratic Formula
\( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)
Examples
\( x^2-4x+3=0 \): 3,1
\( 2x^2+3x-2=0 \): 0.5,-2
\( x^2-2x+5=0 \): \( 1 \pm 2i \)
Vertex of Parabola
\( x = -\frac{b}{2a} \)
Examples
y=x²+4x+1: x=-2
y=2x²-8x+3: x=2
y=ax²+bx+c: x=-b/2a
Factoring Quadratics
\( (x+a)(x+b)=x^2+(a+b)x+ab \)
Examples
x²+5x+6=(x+2)(x+3)
x²-x-6=(x-3)(x+2)
2x²-7x+3=(2x-1)(x-3)
Difference of Squares
\( a^2-b^2=(a+b)(a-b) \)
Examples
x²-9=(x+3)(x-3)
4x²-25=(2x+5)(2x-5)
9y²-16z²=(3y+4z)(3y-4z)
Perfect Square Trinomial
\( a^2+2ab+b^2=(a+b)^2 \)
Examples
x²+6x+9=(x+3)²
y²-10y+25=(y-5)²
a²+4ab+4b²=(a+2b)²
Sum/Diff of Cubes
\( a^3+b^3=(a+b)(a^2-ab+b^2) \)
\( a^3-b^3=(a-b)(a^2+ab+b^2) \)
Examples
x³+8=(x+2)(x²-2x+4)
27y³-1=(3y-1)(9y²+3y+1)
a³-b³=(a-b)(a²+ab+b²)
Discriminant
\( b^2-4ac \)
Examples
x²-4x+3: 4
x²+2x+5: -16
x²-6x+9: 0
Sum/Product of Solutions
Sum: \( -b/a \)
Product: \( c/a \)
Examples
x²-5x+6: sum=5, prod=6
2x²+3x-2: sum=-3/2, prod=-1
x²+4x+4: sum=-4, prod=4

Exponents, Roots, & Polynomials

Product Rule
\( a^m \cdot a^n = a^{m+n} \)
Examples
2²·2³=2?=32
x?·x?=x¹¹
(3x²)³·3x?=3?x¹?
Quotient Rule
\( \frac{a^m}{a^n} = a^{m-n} \)
Examples
5?/5²=5²=25
x?/x³=x²
(2y?)/(4y²)=½y³
Power Rule
\( (a^m)^n = a^{mn} \)
Examples
(2³)²=2?=64
(x²)?=x?
(3a²b³)²=9a?b?
Negative Exponent
\( a^{-n} = \frac{1}{a^n} \)
Examples
4?²=1/16
x?³=1/x³
(2y)?¹=1/(2y)
Fractional Exponents
\( a^{m/n} = \sqrt[n]{a^m} \)
Examples
16^{1/2}=4
8^{2/3}=4
81^{3/4}=27
Zero Exponent
\( a^0 = 1 \) (a?0)
Examples
5?=1
x?=1
(3y²-7x?)?=1
Radical Product
\( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
Examples
?12=2?3
?50=5?2
?18x?=3x²?2
Absolute Value
\( |x| = x \) if \( x \geq 0 \), \( |x| = -x \) if \( x < 0 \)
Examples
|5|=5
|-3|=3
|x-2| is always non-negative

Geometry & Trigonometry

Area of Triangle
\( A = \frac{1}{2}bh \)
Examples
b=6, h=4: 12
b=10, h=7: 35
b=x+2, h=3x: ½(x+2)(3x)
Area of Rectangle
\( A = lw \)
Examples
l=7, w=3: 21
l=12, w=9: 108
l=x, w=x+5: x(x+5)
Area of Parallelogram
\( A = bh \)
Examples
b=8, h=5: 40
b=13, h=4: 52
b=x+2, h=2x: (x+2)(2x)
Area of Trapezoid
\( A = \frac{1}{2}(b_1+b_2)h \)
Examples
b?=6, b?=10, h=4: 32
b?=8, b?=12, h=5: 50
b?=x, b?=2x, h=x: 1.5x²
Area of Circle
\( A = \pi r^2 \)
Examples
r=3: 9?
r=7: 49?
r=x+1: ?(x+1)²
Circumference of Circle
\( C = 2\pi r \)
Examples
r=5: 10?
r=12: 24?
r=x: 2?x
Equation of Circle
\( (x-h)^2 + (y-k)^2 = r^2 \)
Examples
Center(2,-1), r=4: (x-2)²+(y+1)²=16
Center(0,0), r=5: x²+y²=25
Center(h,k), r=r: (x-h)²+(y-k)²=r²
Area of Sector
\( A = \frac{\theta}{360}\pi r^2 \)
Examples
r=6, ?=60°: 6?
r=5, ?=90°: 6.25?
r=x, ?=120°: (1/3)?x²
Arc Length
\( L = \frac{\theta}{360}2\pi r \)
Examples
r=5, ?=90°: 2.5?
r=8, ?=45°: 2?
r=x, ?=180°: ?x
Pythagorean Theorem
\( a^2 + b^2 = c^2 \)
Examples
a=3, b=4: c=5
a=5, b=12: c=13
a=x, b=2x: c=x?5
Special Right Triangles
45-45-90: x,x,x?2
30-60-90: x,x?3,2x
Examples
45-45-90: x,x,x?2
30-60-90: x,x?3,2x
Hypotenuse 10 in 45-45-90: legs=5?2
Volume of Rectangular Prism
\( V = lwh \)
Examples
l=3,w=4,h=5: 60
l=2x,w=x,h=3: 6x²
l=a,w=b,h=c: abc
Volume of Cylinder
\( V = \pi r^2 h \)
Examples
r=2,h=7: 28?
r=5,h=10: 250?
r=x,h=2x: 2?x³
Volume of Sphere
\( V = \frac{4}{3}\pi r^3 \)
Examples
r=3: 36?
r=6: 288?
r=x: (4/3)?x³
Volume of Cone
\( V = \frac{1}{3}\pi r^2 h \)
Examples
r=2,h=9: 12?
r=5,h=6: 50?
r=x,h=x: (1/3)?x³
Volume of Pyramid
\( V = \frac{1}{3}lwh \)
Examples
l=6,w=6,h=10: 120
l=a,w=2a,h=3a: 2a³
l=x,w=y,h=z: (1/3)xyz
Sum of Interior Angles (Polygon)
\( (n-2) \times 180^\circ \)
Examples
Triangle: 180°
Hexagon: 720°
Decagon: 1440°
SOHCAHTOA (Trig Ratios)
\( \sin = \frac{\text{opp}}{\text{hyp}} \), \( \cos = \frac{\text{adj}}{\text{hyp}} \), \( \tan = \frac{\text{opp}}{\text{adj}} \)
Examples
sin(?)=opp/hyp
cos(?)=adj/hyp
tan(?)=opp/adj
Distance = Rate × Time
\( d = rt \)
Examples
r=60, t=2: d=120
r=45, t=1.5: d=67.5
r=x, t=y: d=xy

Statistics, Probability & Data

Mean (Average)
\( \text{Mean} = \frac{\text{Sum}}{\text{Count}} \)
Examples
2,4,6: 4
3,7,8,12: 7.5
x,x+2,x+4: x+2
Median
Middle value when ordered
Examples
3,7,9: 7
2,4,6,8: (4+6)/2=5
x,x+1,x+2,x+3,x+4: x+2
Mode
Most frequent value
Examples
2,3,3,4: 3
1,1,2,3,4,4: 1 and 4
5,5,5,6,7,8: 5
Range
max - min
Examples
2,7,12: 10
4,8,15,20: 16
x,x+5,x+10: 10
Probability
\( P(A) = \frac{\text{favorable}}{\text{total}} \)
Examples
1 red out of 4: 1/4
2 even on die: 3/6=1/2
2 aces in row: 4/52 × 3/51
Probability of A and B
\( P(A \text{ and } B) = P(A) \cdot P(B) \)
Examples
1/2 and 1/3: 1/6
0.4 and 0.5: 0.2
a and b: ab
Fundamental Counting Principle
Total = ways? × ways? × ...
Examples
3 shirts, 2 pants: 6
4 appetizers, 5 entrees: 20
x shirts, y pants, z shoes: xyz
Average Speed
\( \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \)
Examples
60mi in 2hr: 30mph
120mi in 3hr: 40mph
d miles, t hr: d/t

Sequences & Functions

Arithmetic Sequence
\( a_n = a_1 + (n-1)d \)
Examples
a?=2, d=3, n=4: a?=11
a?=5, d=-2, n=6: a?=-5
a?=x, d=2x, n=5: a?=9x
Geometric Sequence
\( a_n = a_1 r^{n-1} \)
Examples
a?=2, r=3, n=3: a?=18
a?=5, r=0.5, n=4: a?=0.625
a?=x, r=2, n=n: a?=x·2^{n-1}
Sum of Arithmetic Series
\( S_n = \frac{n}{2}(a_1 + a_n) \)
Examples
a?=2, a??=20, n=10: S??=110
a?=3, a?=15, n=5: S?=45
a?=x, a?=y, n=n: S?=(n/2)(x+y)
Sum of Geometric Series (finite)
\( S_n = a_1 \frac{1 - r^n}{1 - r} \)
Examples
a?=2, r=3, n=4: S?=80
a?=1, r=2, n=5: S?=31
a?=x, r=y, n=n: S?=x(1-y?)/(1-y)
Exponential Growth/Decay
\( A = P(1 \pm r)^n \)
Examples
$500, 4%/yr, 3yr: 500(1.04)³?562.43
$2000, -10%/yr, 2yr: 1620
$x, r%, n yr: x(1+r/100)?
Function Notation
\( f(x) \)
Examples
f(x)=2x+1, f(3)=7
f(x)=x²-4x, f(2)=-4
f(x)=ax+b, f(y)=ay+b

Desmos SAT Calculator Tips

Graph to Solve: Enter equations directly to find intersections (solutions), roots, or visualize inequalities.
Sliders for Variables: Enter equations with variables (e.g. y = mx + b) and use sliders to see how changing values affects the graph.
Linear Regression: Enter data as a table, then type y_1 ~ mx_1 + b to get the best-fit line for scatterplots.
Plug in Answer Choices: Use tables or sliders to quickly check which answer choices satisfy the equation.
Inequalities: Enter inequalities (like y < 2x+3) to see shaded solution regions.
Function Evaluation: Define functions (e.g., f(x) = 2x^2 - 3) and plug in values directly.
Practice with Official Interface: Use desmos.com/practice and select "College Board" mode.

Quick Reference Table

Formula Name Formula Desmos Usage
Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) Graph two points, draw line, check slope
Slope-Intercept Form \( y = mx + b \) Graph line, adjust m and b with sliders
Point-Slope Form \( y - y_1 = m(x - x_1) \) Graph using sliders for m, \( x_1 \), \( y_1 \)
Midpoint \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) Plot both points, use midpoint formula
Distance Formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) Plot points, use Desmos’s distance tool
Quadratic Formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) Graph quadratic, find x-intercepts
Vertex of Parabola \( x = -\frac{b}{2a} \) Plot vertex, use sliders for a, b
Factoring Quadratics \( (x + a)(x + b) = x^2 + (a+b)x + ab \) Expand/factor using Desmos
Difference of Squares \( a^2 - b^2 = (a+b)(a-b) \) Expand/factor using Desmos
Product Rule (Exponents) \( a^m \cdot a^n = a^{m+n} \) Check with table or direct calculation
Quotient Rule (Exponents) \( \frac{a^m}{a^n} = a^{m-n} \) Check with table or direct calculation
Power Rule (Exponents) \( (a^m)^n = a^{mn} \) Check with table or direct calculation
Area of Triangle \( A = \frac{1}{2}bh \) Plot triangle, use formula for area
Area of Rectangle \( A = lw \) Draw rectangle, use area formula
Area of Circle \( A = \pi r^2 \) Use Desmos’s circle tool for visualization
Circumference of Circle \( C = 2\pi r \) Use Desmos’s circle tool for circumference
Pythagorean Theorem \( a^2 + b^2 = c^2 \) Plot triangle, use distance tool
Special Right Triangles 45-45-90: \( x, x, x\sqrt{2} \)
30-60-90: \( x, x\sqrt{3}, 2x \)		 Label triangle sides, check ratios
Volume of Rectangular Prism \( V = lwh \) Use sliders for l, w, h to visualize volume
Volume of Cylinder \( V = \pi r^2 h \) Use sliders for r, h to visualize volume
Sum of Interior Angles (Polygon) \( (n-2) \times 180^\circ \) Table for different n values
SOHCAHTOA \( \sin = \frac{\text{opp}}{\text{hyp}} \), \( \cos = \frac{\text{adj}}{\text{hyp}} \), \( \tan = \frac{\text{opp}}{\text{adj}} \) Label triangle, use calculator for ratios
Exponential Growth/Decay \( A = P(1 \pm r)^n \) Enter as function, use table
Arithmetic Sequence \( a_n = a_1 + (n-1)d \) Enter as function, use table
Geometric Sequence \( a_n = a_1 r^{n-1} \) Enter as function, use table
Mean (Average) \( \text{Mean} = \frac{\text{Sum}}{\text{Count}} \) Use table, compute sum/count
Median Middle value when ordered Sort values in table
Probability \( P(A) = \frac{\text{favorable}}{\text{total}} \) Use table for counting outcomes
Distance = Rate × Time \( d = rt \) Enter as function, use sliders for r, t

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