Mth1112 Pre Calc With Algebra Formula Sheet

MTH1112 Precalculus with Algebra: A Comprehensive Formula Sheet and Study Guide

Precalculus, often a stepping stone to calculus, requires a solid grasp of foundational algebraic concepts. This comprehensive guide serves as both a formula sheet for MTH1112 (or similar precalculus courses) and a study aid, designed to help you conquer the key concepts and formulas. We'll break down the essential topics, providing explanations and examples to ensure a thorough understanding. Remember, rote memorization isn't enough; true mastery comes from understanding the why behind the formulas.

I. Real Numbers and their Properties

Understanding real numbers and their properties is paramount in precalculus. This section covers the fundamental building blocks of mathematical operations.

A. Number Sets:

• Natural Numbers (ℕ): {1, 2, 3, ...} Positive integers.
• Whole Numbers (𝕎): {0, 1, 2, 3, ...} Natural numbers including zero.
• Integers (ℤ): {..., -3, -2, -1, 0, 1, 2, 3, ...} Positive and negative whole numbers.
• Rational Numbers (ℚ): Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. These include terminating and repeating decimals.
• Irrational Numbers: Numbers that cannot be expressed as a fraction p/q. Examples include π (pi) and √2.
• Real Numbers (ℝ): The union of rational and irrational numbers. This encompasses all numbers on the number line.

B. Properties of Real Numbers:

• Commutative Property: a + b = b + a and a * b = b * a
• Associative Property: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)
• Distributive Property: a(b + c) = ab + ac
• Identity Property: a + 0 = a and a * 1 = a
• Inverse Property: a + (-a) = 0 and a * (1/a) = 1 (where a ≠ 0)

II. Algebraic Expressions and Equations

This section delves into manipulating and solving algebraic expressions and equations, a cornerstone of precalculus.

A. Simplifying Expressions:

• Combining Like Terms: Terms with the same variables raised to the same powers can be added or subtracted. Example: 3x² + 5x² = 8x²
• Distributive Property: Use this to expand expressions like a(b + c).
• Exponents: Remember the rules for exponents:
  • Product Rule: xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾
  • Quotient Rule: xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾
  • Power Rule: (xᵃ)ᵇ = x⁽ᵃ*ᵇ⁾
  • Zero Exponent: x⁰ = 1 (where x ≠ 0)
  • Negative Exponent: x⁻ⁿ = 1/xⁿ
• Radicals: Understand how to simplify radicals and rationalize denominators. Remember that √(a*b) = √a * √b and √(a/b) = √a / √b.

B. Solving Equations:

• Linear Equations: Equations of the form ax + b = c. Solve by isolating the variable.
• Quadratic Equations: Equations of the form ax² + bx + c = 0. Solving methods include:
  • Factoring: If possible, factor the quadratic expression and set each factor to zero.
  • Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
  • Completing the Square: A method to rewrite the quadratic in a perfect square form.
• Systems of Equations: Solve for multiple variables using methods like substitution or elimination.

C. Inequalities:

• Linear Inequalities: Solve similarly to linear equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
• Compound Inequalities: Involve multiple inequalities (e.g., a < x < b).
• Absolute Value Inequalities: Involve absolute value expressions (e.g., |x| < a or |x| > a).

III. Functions

Understanding functions is crucial for success in precalculus and beyond. This section covers function notation, types of functions, and key characteristics.

A. Function Notation:

• f(x): Represents the output of a function f for a given input x.
• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Vertical Line Test: A visual test to determine if a graph represents a function. If any vertical line intersects the graph more than once, it's not a function.

B. Types of Functions:

• Linear Functions: Functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: Functions of the form f(x) = ax² + bx + c, where a, b, and c are constants (a ≠ 0). Their graphs are parabolas.
• Polynomial Functions: Functions that are sums of power functions (e.g., f(x) = x³ + 2x² - x + 1).
• Rational Functions: Functions that are ratios of polynomials (e.g., f(x) = (x² + 1) / (x - 2)).
• Exponential Functions: Functions of the form f(x) = aᵇˣ, where a and b are constants (a > 0, b > 0, and b ≠ 1).
• Logarithmic Functions: The inverse functions of exponential functions.

C. Function Transformations:

Understanding how transformations affect the graph of a function is essential.

• Vertical Shift: f(x) + k (shifts up k units if k > 0, down if k < 0)
• Horizontal Shift: f(x - h) (shifts right h units if h > 0, left if h < 0)
• Vertical Stretch/Compression: cf(x) (stretches vertically if c > 1, compresses if 0 < c < 1)
• Horizontal Stretch/Compression: f(dx) (compresses horizontally if d > 1, stretches if 0 < d < 1)
• Reflection: -f(x) (reflects across the x-axis), f(-x) (reflects across the y-axis)

IV. Trigonometry

Trigonometry is a significant component of precalculus. Mastering these concepts is crucial for further mathematical studies.

A. Basic Trigonometric Functions:

• Sine (sin): Opposite/Hypotenuse
• Cosine (cos): Adjacent/Hypotenuse
• Tangent (tan): Opposite/Adjacent
• Cosecant (csc): 1/sin
• Secant (sec): 1/cos
• Cotangent (cot): 1/tan

B. Trigonometric Identities:

Memorizing and understanding these identities is crucial for simplifying trigonometric expressions and solving trigonometric equations. Key identities include:

• Pythagorean Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
• Sum and Difference Formulas: These formulas allow you to find the sine, cosine, and tangent of the sum or difference of two angles.
• Double-Angle Formulas: These formulas express trigonometric functions of 2θ in terms of trigonometric functions of θ.
• Half-Angle Formulas: These formulas express trigonometric functions of θ/2 in terms of trigonometric functions of θ.

C. Unit Circle:

Understanding the unit circle is vital for quickly determining the values of trigonometric functions for various angles.

V. Sequences and Series

This section introduces sequences and series, laying the groundwork for calculus concepts like limits and infinite series.

A. Sequences:

• Arithmetic Sequence: A sequence where the difference between consecutive terms is constant (common difference).
• Geometric Sequence: A sequence where the ratio between consecutive terms is constant (common ratio).

B. Series:

• Arithmetic Series: The sum of an arithmetic sequence.
• Geometric Series: The sum of a geometric sequence. The formula for an infinite geometric series converges only if the absolute value of the common ratio is less than 1.

VI. Exponential and Logarithmic Functions

These functions are fundamental in numerous applications and form the basis for understanding growth and decay models.

A. Exponential Functions:

• Growth: f(x) = a(1 + r)ˣ where 'a' is the initial value, 'r' is the growth rate, and 'x' is time.
• Decay: f(x) = a(1 - r)ˣ where 'a' is the initial value, 'r' is the decay rate, and 'x' is time.

B. Logarithmic Functions:

• Logarithms: The inverse of exponential functions. Understanding the properties of logarithms is critical for solving logarithmic equations and simplifying expressions. Key properties include:
  • Product Rule: logₐ(xy) = logₐx + logₐy
  • Quotient Rule: logₐ(x/y) = logₐx - logₐy
  • Power Rule: logₐ(xⁿ) = n logₐx
  • Change of Base Formula: logₐx = (logₓx / logₐx)

This comprehensive guide provides a solid foundation for your MTH1112 precalculus course. Remember to practice regularly, work through examples, and seek help when needed. Understanding the underlying concepts is as important as memorizing the formulas. Good luck!