SAT Math Study Guide

Linear Equations

• Definition: An equation represented by a straight line on a graph.
• Slope-intercept form: y = mx + b
  • m is the slope
  • b is the y-intercept

Types of Slopes

• Positive slope: Line is uphill (m > 0)
• Negative slope: Line is downhill (m < 0)
• Zero slope: Line is horizontal (m = 0)
• Undefined slope: Line is vertical (m is undefined)

Memorization Aid

• VUX HOY:
  • V: Vertical line, U: Undefined, X: x = #
  • H: Horizontal, O: slope = 0, Y: y = #
• Finding slope: m = (Y2 - Y1) / (X2 - X1)

Quadratic Equations

• Discriminant: D = b² - 4ac
  • D < 0: 2 imaginary roots
  • D > 0: 2 real roots
  • D = 0: 1 real root repeated
• Vertex Form: y = a(x - h)² + k
  • (h, k) is the vertex
  • Steps to find a:
    1. Find vertex
    2. Find a point through the line
    3. Substitute h, k, x, y into equation
    4. Solve for a

Graphing Terms

• Maximum/Minimum: Highest/Lowest y value
• Domain: All x values
• Range: All y values
• Intercepts:
  • x-intercept: Where f(x) = 0
  • y-intercept: Where f(0) = #

Factor Formulas

• Difference of Squares: a² - b² = (a + b)(a - b)
• Factor by Grouping: E.g., 2x³ - 10x² + 3x - 15 = (x - 5)(2x² + 3)
• Perfect Binomial Squared:
  • (x + y)² = x² + 2xy + y²
  • (x - y)² = x² - 2xy + y²

Complex Numbers

• Form: a + bi
  • a: Real part, b: Imaginary part
• Powers of i:
  • i = √-1, i² = -1, i³ = -i, i⁴ = 1

Absolute Value

• Steps:
  1. Isolate absolute value
  2. Create positive/negative equations
  3. Solve for variable
  4. Check for extraneous solutions

Graphing

• Formula: y = a |x - h| + k
  • (h, k) is vertex
  • Axis of symmetry: x = h

Exponents

• Product Rule: a^m * a^n = a^(m+n)
• Quotient Rule: a^m / a^n = a^(m-n)
• Power Rule: (a^m)^n = a^(mn)
• Zero Rule: a^0 = 1
• Negative Exponent: a^-n = 1/a^n*

Radicals

• Definition: Expression with a root
• To remove a radical: Square both sides
• Rationalize denominator by multiplying top and bottom with same term

Systems of Equations

• Methods: Substitution (Recommended for SAT), Elimination
• Solutions:
  • One Solution: Intersect at one point
  • No Solution: Parallel lines
  • Infinite Solutions: Same line

Exponential Equations

• Formula: y = ab^x
  • a ≠ 0, b > 0, b ≠ 1

Inequalities

• In Word Problems:
  • Greater than: Above, more than
  • Greater than or equal to: At least
  • Less than: Below, fewer than
  • Less than or equal to: At most

Graphing

• Greater than: Dashed line, shade above
• Greater than or equal to: Solid line, shade above

Function Notation

• Solve by plugging x into f(x)
• E.g., f(x) = x² + 2x + 4, f(-4) = 12

Geometry

• Special Right Triangles: 45°-45°-90°, 30°-60°-90°
• Pythagorean Theorem: a² + b² = c²
• Types of Angles:
  • Complementary: Sum is 90°
  • Supplementary: Sum is 180°

Circle Equations

• Formula: (x - h)² + (y - k)² = r²
  • (h, k) is center, r is radius

Statistics

• Mean, Median, Mode, Range
• Standard Deviation
• Correlation (r)

Problem Solving

• Ratio: a : b
• Proportion: a/b = c/d
• Percentage calculations

Probability

• Formula: Desired outcome / Total outcomes

Trigonometry

• SOH CAH TOA
• Radians to Degrees: (π / 180) * degrees*

Additional Resources

• Practice SAT Math Questions: Khan Academy, College Board
• Recommended SAT Books: College Board, Princeton Review, College Panda, Barrons

This guide consolidates key math concepts for SAT preparation, offering essential formulas, rules, and problem-solving techniques.