Mth1112 Pre Calc With Algebra Formula Sheet Troy

MTH1112 Precalculus with Algebra Formula Sheet: Your Troy University Success Guide

Precalculus can feel like a daunting task, especially when faced with a mountain of formulas. This comprehensive guide serves as your ultimate MTH1112 Precalculus with Algebra formula sheet, designed specifically for Troy University students. We'll break down key concepts, provide example problems, and offer strategies for mastering this crucial course. This isn't just a list; it's your roadmap to success!

I. Essential Algebraic Concepts: The Foundation of Precalculus

Before diving into the more advanced precalculus concepts, let's solidify our understanding of fundamental algebra. Many students find that a strong algebraic foundation is the key to unlocking success in precalculus.

A. Real Numbers and Their Properties

• Real Numbers: The set of all rational (fractions) and irrational (non-repeating, non-terminating decimals) numbers.
• Properties: Commutative (a + b = b + a, ab = ba), Associative (a + (b + c) = (a + b) + c, a(bc) = (ab)c), Distributive (a(b + c) = ab + ac), Identity (a + 0 = a, a * 1 = a), Inverse (a + (-a) = 0, a * (1/a) = 1, a ≠ 0).
• Order of Operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Remember this acronym to avoid common calculation errors.

Example: Simplify the expression: 3(2x + 5) - 4x + 10

Using the distributive property and then combining like terms:

6x + 15 - 4x + 10 = 2x + 25

B. Solving Equations and Inequalities

• Linear Equations: Equations of the form ax + b = c. Solve by isolating the variable (x).
• Quadratic Equations: Equations of the form ax² + bx + c = 0. Solve using factoring, the quadratic formula (x = [-b ± √(b² - 4ac)] / 2a), or completing the square.
• Inequalities: Similar to equations but use inequality symbols (<, >, ≤, ≥). Remember to flip the inequality sign when multiplying or dividing by a negative number.

Example: Solve the quadratic equation: x² - 5x + 6 = 0

Factoring: (x - 2)(x - 3) = 0, so x = 2 or x = 3

Example: Solve the inequality: 2x - 4 > 6

Add 4 to both sides: 2x > 10

Divide by 2: x > 5

C. Exponents and Radicals

• Exponent Rules: xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾, xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾, (xᵃ)ᵇ = x⁽ᵃᵇ⁾, x⁰ = 1 (x ≠ 0), x⁻ᵃ = 1/xᵃ, √x = x¹/²

• Radicals: The nth root of a number x is denoted as √ⁿx or x^(1/n). Remember that simplifying radicals often involves factoring out perfect squares (or cubes, etc.).

Example: Simplify: (x³)² * x⁻¹

(x³)² * x⁻¹ = x⁶ * x⁻¹ = x⁵

Example: Simplify √75

√75 = √(25 * 3) = √25 * √3 = 5√3

D. Functions and their Graphs

• Function Notation: f(x) represents the output of a function f when the input is x.
• Domain and Range: The set of all possible input values (x-values) and output values (y-values), respectively.
• Graphing Functions: Understanding how to plot points and identify key features like intercepts, asymptotes, and symmetry is crucial.

Example: Find the domain and range of f(x) = √(x - 2)

The domain is x ≥ 2 (since you can't take the square root of a negative number). The range is y ≥ 0.

II. Core Precalculus Topics: Building Upon the Foundation

Now that we've refreshed our algebra skills, let's move into the core concepts of precalculus.

A. Polynomial Functions

• Definition: A function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer and aₙ ≠ 0.
• Degree: The highest power of x in the polynomial.
• Roots/Zeros: The x-values where f(x) = 0. These can be found by factoring, using the quadratic formula (for quadratic polynomials), or employing numerical methods for higher-degree polynomials.

B. Rational Functions

• Definition: A function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
• Vertical Asymptotes: Occur where the denominator is equal to zero and the numerator is not zero.
• Horizontal Asymptotes: Determine the behavior of the function as x approaches positive or negative infinity. This depends on the degrees of P(x) and Q(x).
• Holes: Occur when there's a common factor in the numerator and denominator that cancels out.

C. Exponential and Logarithmic Functions

• Exponential Functions: Functions of the form f(x) = aᵇˣ, where a > 0 and a ≠ 1.
• Logarithmic Functions: The inverse of exponential functions. If y = aˣ, then x = logₐy.
• Properties of Logarithms: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx - logₐy, logₐ(xⁿ) = n logₐx, logₐa = 1, logₐ1 = 0

Example: Solve for x: 2ˣ = 8

Taking the logarithm base 2 of both sides: log₂(2ˣ) = log₂8 => x = 3

D. Trigonometric Functions

• Unit Circle: Essential for understanding the values of trigonometric functions at key angles (0°, 30°, 45°, 60°, 90°, etc.).
• Basic Trigonometric Functions: sin θ, cos θ, tan θ, csc θ, sec θ, cot θ.
• Trigonometric Identities: Relationships between trigonometric functions (e.g., sin²θ + cos²θ = 1, tan θ = sin θ / cos θ). Mastering these identities is crucial for simplifying expressions and solving trigonometric equations.

E. Trigonometric Graphs

• Graphs of Sine, Cosine, and Tangent: Understanding the amplitude, period, phase shift, and vertical shift of these graphs is key.
• Inverse Trigonometric Functions: arcsin x, arccos x, arctan x. These functions return the angle whose sine, cosine, or tangent is x.

F. Analytic Geometry: Conics

• Parabolas: Graphs of quadratic equations.
• Circles: Graphs of equations of the form (x - h)² + (y - k)² = r².
• Ellipses: Graphs of equations of the form (x - h)²/a² + (y - k)²/b² = 1 (or vice versa).
• Hyperbolas: Graphs of equations of the form (x - h)²/a² - (y - k)²/b² = 1 (or vice versa).

III. Strategies for Success in MTH1112

This formula sheet is a valuable resource, but it's only part of the equation for success in MTH1112. Here are some additional strategies:

• Attend every class: Active participation enhances understanding and provides opportunities to ask questions.
• Do your homework regularly: Consistent practice solidifies your understanding of concepts.
• Form study groups: Collaborating with peers helps to identify weaknesses and share knowledge.
• Utilize office hours: Don't hesitate to seek help from your professor or teaching assistant.
• Practice, practice, practice: The more you practice, the more confident you'll become. Work through plenty of problems from your textbook and other resources.
• Seek additional help if needed: Troy University likely offers tutoring services or supplemental instruction. Take advantage of these resources.
• Master the fundamental concepts: A strong foundation in algebra is crucial for success in precalculus. Review these areas if you feel weak.
• Focus on understanding, not just memorization: While memorizing some formulas is necessary, a deeper understanding of the underlying concepts will lead to greater success.

This comprehensive guide, while not exhaustive, serves as a robust foundation for your MTH1112 journey. Remember, consistent effort and a strategic approach are key to conquering precalculus. Good luck!