1. Fundamental Algebraic Concepts
Understanding the basic principles of algebra is crucial for success in college algebra. Here are some fundamental concepts you should master:
1.1 Variables and Constants
- Variable: A symbol (usually a letter) that represents an unknown quantity (e.g., x, y).
- Constant: A fixed value that does not change (e.g., 5, -3.14).
1.2 Expressions and Equations
- Expression: A combination of variables, constants, and operators (e.g., 3x + 5).
- Equation: A statement that two expressions are equal, typically containing an equals sign (e.g., 2x + 3 = 7).
1.3 Order of Operations
To simplify expressions and solve equations, remember the order of operations, often abbreviated as PEMDAS:
1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)
2. Linear Equations
Linear equations are foundational in algebra. They represent straight lines when graphed and can be expressed in various forms.
2.1 Standard Form
The standard form of a linear equation is given by:
\[ Ax + By = C \]
Where:
- \( A \), \( B \), and \( C \) are constants.
- \( A \) and \( B \) cannot both be zero.
2.2 Slope-Intercept Form
The slope-intercept form is expressed as:
\[ y = mx + b \]
Where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept (the point where the line crosses the y-axis).
2.3 Point-Slope Form
Point-slope form is useful for writing the equation of a line when you know one point and the slope:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( (x_1, y_1) \) is a point on the line.
3. Quadratic Equations
Quadratic equations are polynomial equations of degree two and can be represented as:
\[ ax^2 + bx + c = 0 \]
Where \( a \), \( b \), and \( c \) are constants.
3.1 Factoring Quadratics
Quadratics can often be factored into the form:
\[ (px + q)(rx + s) = 0 \]
To solve, set each factor equal to zero:
1. \( px + q = 0 \)
2. \( rx + s = 0 \)
3.2 Quadratic Formula
When factoring is not possible, use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where:
- The term \( b^2 - 4ac \) is called the discriminant and indicates the nature of the roots.
3.3 Completing the Square
Completing the square involves rewriting a quadratic equation in the form:
\[ (x - p)^2 = q \]
To solve:
1. Isolate the \( x^2 \) and \( x \) terms.
2. Add and subtract the square of half the coefficient of \( x \).
3. Solve for \( x \).
4. Functions and Graphs
Functions are vital in algebra, representing relationships between variables.
4.1 Definition of a Function
A function is a relation that assigns exactly one output for each input. If \( f(x) \) represents a function, then:
- \( f(a) = b \) indicates that \( a \) is mapped to \( b \).
4.2 Types of Functions
- Linear Function: \( f(x) = mx + b \)
- Quadratic Function: \( f(x) = ax^2 + bx + c \)
- Exponential Function: \( f(x) = a(b^x) \)
- Logarithmic Function: \( f(x) = \log_b(x) \)
4.3 Graphing Functions
To graph functions:
1. Identify key features (intercepts, slope).
2. Create a table of values.
3. Plot points on a coordinate plane.
4. Connect points smoothly for curves.
5. Systems of Equations
Systems of equations involve solving for multiple variables simultaneously.
5.1 Solving Systems by Substitution
1. Solve one equation for one variable.
2. Substitute that expression into the other equation.
3. Solve for the remaining variable.
5.2 Solving Systems by Elimination
1. Align equations vertically.
2. Add or subtract equations to eliminate one variable.
3. Solve for the remaining variable.
5.3 Graphical Method
1. Graph both equations on the same coordinate plane.
2. Identify the point(s) of intersection, which are the solutions.
6. Exponents and Radicals
Understanding exponents and radicals is critical for manipulating algebraic expressions.
6.1 Laws of Exponents
1. \( x^a \cdot x^b = x^{a+b} \)
2. \( \frac{x^a}{x^b} = x^{a-b} \)
3. \( (x^a)^b = x^{ab} \)
4. \( x^{-a} = \frac{1}{x^a} \)
6.2 Simplifying Radicals
- To simplify \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
- Rationalize denominators by multiplying by the conjugate.
7. Polynomials
Polynomials are expressions consisting of variables raised to whole number powers.
7.1 Polynomial Operations
1. Addition/Subtraction: Combine like terms.
2. Multiplication: Use the distributive property or FOIL for binomials.
3. Division: Use long division or synthetic division.
7.2 The Remainder and Factor Theorems
- Remainder Theorem: If a polynomial \( f(x) \) is divided by \( x - c \), the remainder is \( f(c) \).
- Factor Theorem: \( x - c \) is a factor of \( f(x) \) if \( f(c) = 0 \).
8. Additional Topics
8.1 Inequalities
Inequalities express a relationship where one side is greater or less than the other.
- Linear Inequality: \( ax + b < c \)
- Solving Inequalities: Similar to equations, but reverse the inequality sign when multiplying/dividing by a negative.
8.2 Absolute Value Equations
An absolute value equation is of the form:
\[ |x| = a \]
Solutions are:
1. \( x = a \)
2. \( x = -a \)
8.3 Sequences and Series
- Arithmetic Sequence: A sequence with a common difference \( d \).
- Geometric Sequence: A sequence with a common ratio \( r \).
The sum of the first \( n \) terms of an arithmetic sequence can be calculated as:
\[ S_n = \frac{n}{2} (a + l) \]
Where \( a \) is the first term, \( l \) is the last term.
Conclusion
The college algebra formula cheat sheet condenses essential algebraic concepts and formulas into a single, accessible guide. Mastery of these principles allows students to tackle complex problems with confidence, laying a strong foundation for advanced mathematical studies. Whether you're solving equations, working with functions, or grappling with polynomials, this cheat sheet will serve as a reliable companion throughout your college algebra journey. Consistent practice and application of these formulas will not only enhance your understanding but also boost your performance in exams and coursework.
Frequently Asked Questions
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