SAT Math Formula Sheet & Desmos Power Guide

Algebra & Linear Equations

• Slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
  Example 1: (1, 2) and (3, 6): \( m = \frac{6-2}{3-1} = 2 \)
  Example 2: (-2, 5) and (4, -1): \( m = \frac{-1-5}{4-(-2)} = -1 \)
  Example 3: (a, 2a) and (a+3, 5a): \( m = \frac{5a-2a}{a+3-a} = a \)
• Slope-Intercept Form: \( y = mx + b \)
  Example 1: Slope 2, y-intercept 1: \( y = 2x + 1 \)
  Example 2: Through (0, -3) with slope -4: \( y = -4x - 3 \)
  Example 3: Through (2, 5) and (6, 13): \( m = 2, b = 1 \Rightarrow y = 2x + 1 \)
• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
  Example 1: Slope 3, through (1, 2): \( y - 2 = 3(x - 1) \)
  Example 2: Slope -2, through (4, -5): \( y + 5 = -2(x - 4) \)
  Example 3: Slope \( m \), through (h, k): \( y - k = m(x - h) \)
• Midpoint: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
  Example 1: (2, 4) and (4, 8): \( (3, 6) \)
  Example 2: (-3, 7) and (5, -1): \( (1, 3) \)
  Example 3: (a, b) and (c, d): \( \left( \frac{a+c}{2}, \frac{b+d}{2} \right) \)
• Distance: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
  Example 1: (0, 0) to (3, 4): \( d = 5 \)
  Example 2: (1, 2) to (4, 6): \( d = 5 \)
  Example 3: (a, b) to (a+3, b+4): \( d = 5 \)
• Parallel Lines: Equal slopes
  Example 1: \( y = 2x + 1 \) and \( y = 2x - 3 \) are parallel.
  Example 2: \( 3x - y = 7 \) and \( 6x - 2y = 4 \) are parallel.
  Example 3: If \( m = 5 \), all lines with \( m = 5 \) are parallel.
• Perpendicular Lines: Slopes are negative reciprocals (\( m_1 \cdot m_2 = -1 \))
  Example 1: \( m_1 = 2, m_2 = -\frac{1}{2} \).
  Example 2: \( y = 3x + 1 \) and \( y = -\frac{1}{3}x + 2 \) are perpendicular.
  Example 3: \( y = mx + b \) and \( y = -\frac{1}{m}x + c \) are perpendicular.
• Standard Form: \( Ax + By = C \)
  Example 1: \( 2x + 3y = 6 \)
  Example 2: \( 5x - y = 10 \)
  Example 3: \( 4x + 0y = 8 \) (vertical line)

Quadratics & Polynomials

• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  Example 1: \( x^2 - 4x + 3 = 0 \) ? \( x = 3, 1 \)
  Example 2: \( 2x^2 + 3x - 2 = 0 \) ? \( x = 0.5, -2 \)
  Example 3: \( x^2 - 2x + 5 = 0 \) ? \( x = 1 \pm 2i \)
• Vertex of Parabola: \( x = -\frac{b}{2a} \)
  Example 1: \( y = x^2 + 4x + 1 \Rightarrow x = -2 \)
  Example 2: \( y = 2x^2 - 8x + 3 \Rightarrow x = 2 \)
  Example 3: \( y = ax^2 + bx + c \Rightarrow x = -\frac{b}{2a} \)
• Factoring Quadratics: \( (x + a)(x + b) = x^2 + (a + b)x + ab \)
  Example 1: \( x^2 + 5x + 6 = (x+2)(x+3) \)
  Example 2: \( x^2 - x - 6 = (x-3)(x+2) \)
  Example 3: \( 2x^2 - 7x + 3 = (2x-1)(x-3) \)
• Difference of Squares: \( a^2 - b^2 = (a + b)(a - b) \)
  Example 1: \( x^2 - 9 = (x+3)(x-3) \)
  Example 2: \( 4x^2 - 25 = (2x+5)(2x-5) \)
  Example 3: \( 9y^2 - 16z^2 = (3y+4z)(3y-4z) \)
• Perfect Square Trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \)
  Example 1: \( x^2 + 6x + 9 = (x+3)^2 \)
  Example 2: \( y^2 - 10y + 25 = (y-5)^2 \)
  Example 3: \( a^2 + 4ab + 4b^2 = (a+2b)^2 \)
• Sum and Difference of Cubes:
  Example 1: \( x^3 + 8 = (x+2)(x^2-2x+4) \)
  Example 2: \( 27y^3 - 1 = (3y-1)(9y^2+3y+1) \)
  Example 3: \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \)
• Discriminant: \( b^2 - 4ac \)
  Example 1: \( x^2 - 4x + 3 = 0 \Rightarrow 16 - 12 = 4 \) (2 real roots)
  Example 2: \( x^2 + 2x + 5 = 0 \Rightarrow 4 - 20 = -16 \) (2 complex roots)
  Example 3: \( x^2 - 6x + 9 = 0 \Rightarrow 36 - 36 = 0 \) (1 real root)
• Sum of Solutions: For \( ax^2+bx+c=0 \), sum is \( -\frac{b}{a} \)
  Example 1: \( x^2-5x+6=0 \): sum is 5
  Example 2: \( 2x^2+3x-2=0 \): sum is \(-\frac{3}{2}\)
  Example 3: \( x^2+4x+4=0 \): sum is -4
• Product of Solutions: For \( ax^2+bx+c=0 \), product is \( \frac{c}{a} \)
  Example 1: \( x^2-5x+6=0 \): product is 6
  Example 2: \( 2x^2+3x-2=0 \): product is \(-1\)
  Example 3: \( x^2+4x+4=0 \): product is 4

Exponents, Roots, & Polynomials

• Product Rule: \( a^m \cdot a^n = a^{m+n} \)
  Example 1: \( 2^2 \cdot 2^3 = 2^5 = 32 \)
  Example 2: \( x^4 \cdot x^7 = x^{11} \)
  Example 3: \( (3x^2)^3 \cdot 3x^4 = 3^4 x^{10} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
  Example 1: \( 5^6 / 5^2 = 5^4 = 625 \)
  Example 2: \( x^8 / x^3 = x^5 \)
  Example 3: \( (2y^7)/(4y^2) = (1/2)y^5 \)
• Power Rule: \( (a^m)^n = a^{mn} \)
  Example 1: \( (2^3)^2 = 2^6 = 64 \)
  Example 2: \( (x^2)^5 = x^{10} \)
  Example 3: \( (3a^2b^3)^2 = 9a^4b^6 \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
  Example 1: \( 4^{-2} = 1/16 \)
  Example 2: \( x^{-3} = 1/x^3 \)
  Example 3: \( (2y)^{-1} = 1/(2y) \)
• Fractional Exponents: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)
  Example 1: \( 16^{1/2} = 4 \)
  Example 2: \( 8^{2/3} = 4 \)
  Example 3: \( 81^{3/4} = 27 \)
• Zero Exponent: \( a^0 = 1 \) (for \( a \neq 0 \))
  Example 1: \( 5^0 = 1 \)
  Example 2: \( x^0 = 1 \)
  Example 3: \( (3y^2 - 7x^4)^0 = 1 \)
• Radical Product: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
  Example 1: \( \sqrt{12} = 2\sqrt{3} \)
  Example 2: \( \sqrt{50} = 5\sqrt{2} \)
  Example 3: \( \sqrt{18x^4} = 3x^2\sqrt{2} \)
• Absolute Value: \( |x| = x \) if \( x \geq 0 \), \( |x| = -x \) if \( x < 0 \)
  Example 1: \( |5| = 5 \)
  Example 2: \( |-3| = 3 \)
  Example 3: \( |x-2| \) is always non-negative

Geometry & Trigonometry

• Area of Triangle: \( A = \frac{1}{2}bh \)
  Example 1: \( b = 6, h = 4 \Rightarrow A = 12 \)
  Example 2: \( b = 10, h = 7 \Rightarrow A = 35 \)
  Example 3: \( b = x+2, h = 3x \Rightarrow A = \frac{1}{2}(x+2)(3x) \)
• Area of Rectangle: \( A = lw \)
  Example 1: \( l = 7, w = 3 \Rightarrow A = 21 \)
  Example 2: \( l = 12, w = 9 \Rightarrow A = 108 \)
  Example 3: \( l = x, w = x+5 \Rightarrow A = x(x+5) \)
• Area of Parallelogram: \( A = bh \)
  Example 1: \( b = 8, h = 5 \Rightarrow A = 40 \)
  Example 2: \( b = 13, h = 4 \Rightarrow A = 52 \)
  Example 3: \( b = x+2, h = 2x \Rightarrow A = (x+2)(2x) \)
• Area of Trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \)
  Example 1: \( b_1 = 6, b_2 = 10, h = 4 \Rightarrow A = 32 \)
  Example 2: \( b_1 = 8, b_2 = 12, h = 5 \Rightarrow A = 50 \)
  Example 3: \( b_1 = x, b_2 = 2x, h = x \Rightarrow A = \frac{3x^2}{2} \)
• Area of Circle: \( A = \pi r^2 \)
  Example 1: \( r = 3 \Rightarrow A = 9\pi \)
  Example 2: \( r = 7 \Rightarrow A = 49\pi \)
  Example 3: \( r = x+1 \Rightarrow A = \pi(x+1)^2 \)
• Circumference of Circle: \( C = 2\pi r \)
  Example 1: \( r = 5 \Rightarrow C = 10\pi \)
  Example 2: \( r = 12 \Rightarrow C = 24\pi \)
  Example 3: \( r = x \Rightarrow C = 2\pi x \)
• Equation of Circle: \( (x-h)^2 + (y-k)^2 = r^2 \)
  Example 1: Center (2, -1), radius 4: \( (x-2)^2 + (y+1)^2 = 16 \)
  Example 2: Center (0,0), radius 5: \( x^2 + y^2 = 25 \)
  Example 3: Center (h, k), radius r: \( (x-h)^2 + (y-k)^2 = r^2 \)
• Area of Sector: \( A = \frac{\theta}{360} \times \pi r^2 \)
  Example 1: \( r = 6, \theta = 60^\circ \Rightarrow 6\pi \)
  Example 2: \( r = 5, \theta = 90^\circ \Rightarrow 6.25\pi \)
  Example 3: \( r = x, \theta = 120^\circ \Rightarrow \frac{1}{3}\pi x^2 \)
• Arc Length: \( L = \frac{\theta}{360} \times 2\pi r \)
  Example 1: \( r = 5, \theta = 90^\circ \Rightarrow 2.5\pi \)
  Example 2: \( r = 8, \theta = 45^\circ \Rightarrow 2\pi \)
  Example 3: \( r = x, \theta = 180^\circ \Rightarrow \pi x \)
• Pythagorean Theorem: \( a^2 + b^2 = c^2 \)
  Example 1: \( a=3, b=4 \Rightarrow c=5 \)
  Example 2: \( a=5, b=12 \Rightarrow c=13 \)
  Example 3: \( a=x, b=2x \Rightarrow c=x\sqrt{5} \)
• Special Right Triangles:
  Example 1: 45-45-90: sides \( x, x, x\sqrt{2} \)
  Example 2: 30-60-90: sides \( x, x\sqrt{3}, 2x \)
  Example 3: If hypotenuse is 10 in a 45-45-90, each leg is \( 5\sqrt{2} \)
• Volume of Rectangular Prism: \( V = lwh \)
  Example 1: \( l = 3, w = 4, h = 5 \Rightarrow V = 60 \)
  Example 2: \( l = 2x, w = x, h = 3 \Rightarrow V = 6x^2 \)
  Example 3: \( l = a, w = b, h = c \Rightarrow V = abc \)
• Volume of Cylinder: \( V = \pi r^2 h \)
  Example 1: \( r = 2, h = 7 \Rightarrow V = 28\pi \)
  Example 2: \( r = 5, h = 10 \Rightarrow V = 250\pi \)
  Example 3: \( r = x, h = 2x \Rightarrow V = 2\pi x^3 \)
• Volume of Sphere: \( V = \frac{4}{3}\pi r^3 \)
  Example 1: \( r = 3 \Rightarrow V = 36\pi \)
  Example 2: \( r = 6 \Rightarrow V = 288\pi \)
  Example 3: \( r = x \Rightarrow V = \frac{4}{3}\pi x^3 \)
• Volume of Cone: \( V = \frac{1}{3}\pi r^2 h \)
  Example 1: \( r = 2, h = 9 \Rightarrow V = 12\pi \)
  Example 2: \( r = 5, h = 6 \Rightarrow V = 50\pi \)
  Example 3: \( r = x, h = x \Rightarrow V = \frac{1}{3}\pi x^3 \)
• Volume of Pyramid: \( V = \frac{1}{3}lwh \)
  Example 1: \( l = 6, w = 6, h = 10 \Rightarrow V = 120 \)
  Example 2: \( l = a, w = 2a, h = 3a \Rightarrow V = 2a^3 \)
  Example 3: \( l = x, w = y, h = z \Rightarrow V = \frac{1}{3}xyz \)
• Sum of Interior Angles (Polygon): \( (n-2) \times 180^\circ \)
  Example 1: Triangle: \( (3-2)\times180 = 180^\circ \)
  Example 2: Hexagon: \( (6-2)\times180 = 720^\circ \)
  Example 3: Decagon: \( (10-2)\times180 = 1440^\circ \)
• SOHCAHTOA (Trig Ratios):
  Example 1: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
  Example 2: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
  Example 3: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• Distance = Rate × Time: \( d = rt \)
  Example 1: \( r = 60, t = 2 \rightarrow d = 120 \)
  Example 2: \( r = 45, t = 1.5 \rightarrow d = 67.5 \)
  Example 3: \( r = x, t = y \rightarrow d = xy \)

Statistics, Probability & Data

• Mean (Average): \( \text{Mean} = \frac{\text{Sum of items}}{\text{Number of items}} \)
  Example 1: 2, 4, 6: Mean = 4
  Example 2: 3, 7, 8, 12: Mean = 7.5
  Example 3: \( x, x+2, x+4 \): Mean = \( x+2 \)
• Median: Middle value when data is ordered
  Example 1: 3, 7, 9 ? Median is 7
  Example 2: 2, 4, 6, 8 ? Median is (4+6)/2 = 5
  Example 3: \( x, x+1, x+2, x+3, x+4 \): Median is \( x+2 \)
• Mode: Most frequent value
  Example 1: 2, 3, 3, 4 ? Mode is 3
  Example 2: 1, 1, 2, 3, 4, 4 ? Modes are 1 and 4
  Example 3: 5, 5, 5, 6, 7, 8 ? Mode is 5
• Range: \( \text{Max} - \text{Min} \)
  Example 1: 2, 7, 12 ? Range = 10
  Example 2: 4, 8, 15, 20 ? Range = 16
  Example 3: \( x, x+5, x+10 \): Range = 10
• Probability: \( P(A) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} \)
  Example 1: 1 red out of 4 marbles: \( 1/4 \)
  Example 2: 2 even numbers on a die: \( 3/6 = 1/2 \)
  Example 3: Drawing 2 aces in a row: \( \frac{4}{52} \times \frac{3}{51} \)
• Probability of A and B (independent): \( P(A \text{ and } B) = P(A) \cdot P(B) \)
  Example 1: P(A)=1/2, P(B)=1/3 ? 1/6
  Example 2: P(A)=0.4, P(B)=0.5 ? 0.2
  Example 3: P(A)=a, P(B)=b ? ab
• Fundamental Counting Principle: If one event can occur in N ways and another in M ways, both can occur in \( N \times M \) ways.
  Example 1: 3 shirts, 2 pants: \( 3 \times 2 = 6 \)
  Example 2: 4 appetizers, 5 entrees: \( 20 \)
  Example 3: \( x \) shirts, \( y \) pants, \( z \) shoes: \( xyz \)
• Average Speed: \( \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \)
  Example 1: 60 miles in 2 hours: 30 mph
  Example 2: 120 miles in 3 hours: 40 mph
  Example 3: d miles in t hours: \( \frac{d}{t} \)

Sequences & Functions

• Arithmetic Sequence: \( a_n = a_1 + (n-1)d \)
  Example 1: \( a_1=2, d=3, n=4 \Rightarrow a_4=11 \)
  Example 2: \( a_1=5, d=-2, n=6 \Rightarrow a_6=-5 \)
  Example 3: \( a_1=x, d=2x, n=5 \Rightarrow a_5=9x \)
• Geometric Sequence: \( a_n = a_1 r^{n-1} \)
  Example 1: \( a_1=2, r=3, n=3 \Rightarrow a_3=18 \)
  Example 2: \( a_1=5, r=0.5, n=4 \Rightarrow a_4=0.625 \)
  Example 3: \( a_1=x, r=2, n=n \Rightarrow a_n=x \cdot 2^{n-1} \)
• Sum of Arithmetic Series: \( S_n = \frac{n}{2}(a_1 + a_n) \)
  Example 1: \( a_1=2, a_{10}=20, n=10 \Rightarrow S_{10}=110 \)
  Example 2: \( a_1=3, a_{5}=15, n=5 \Rightarrow S_5=45 \)
  Example 3: \( a_1=x, a_n=y, n=n \Rightarrow S_n=\frac{n}{2}(x+y) \)
• Sum of Geometric Series (finite): \( S_n = a_1 \frac{1 - r^n}{1 - r} \) for \( r \neq 1 \)
  Example 1: \( a_1=2, r=3, n=4 \Rightarrow S_4=80 \)
  Example 2: \( a_1=1, r=2, n=5 \Rightarrow S_5=31 \)
  Example 3: \( a_1=x, r=y, n=n \Rightarrow S_n=x\frac{1-y^n}{1-y} \)
• Exponential Growth/Decay: \( A = P(1 \pm r)^n \)
  Example 1: $500 grows 4% per year for 3 years: \( A = 500(1.04)^3 \approx 562.43 \)
  Example 2: $2000 decays 10% per year for 2 years: \( A = 2000(0.9)^2 = 1620 \)
  Example 3: $x grows r% for n years: \( A = x(1+\frac{r}{100})^n \)
• Function Notation: \( f(x) \)
  Example 1: \( f(x) = 2x + 1, f(3) = 7 \)
  Example 2: \( f(x) = x^2 - 4x, f(2) = -4 \)
  Example 3: \( f(x) = ax+b, f(y) = ay+b \)

Desmos SAT Calculator Tips

Quick Reference Table

Test Day Tips

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