Equations and Inequalities

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, solving equations is a fundamental skill, including linear, quadratic, and rational equations.

• Linear Equation: An equation of the form .

• Quadratic Equation: An equation of the form .

• Rational Equation: An equation involving fractions whose numerators and/or denominators contain a variable.

• Example: Solve .

Inequalities

Inequalities compare two expressions and use symbols such as <, >, ≤, or ≥. Solutions are often expressed as intervals.

• Example: Solve .

  • Solution:

Functions and Graphs

Definition and Properties of Functions

A function is a relation that assigns each input exactly one output. The domain is the set of all possible inputs, and the range is the set of all possible outputs.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

• Example: For , domain is , range is .

Graphing Functions

Graphs visually represent functions. Key features include intercepts, intervals of increase/decrease, and relative extrema.

• Intercepts: Points where the graph crosses the axes.

• Increasing/Decreasing Intervals: Where the function rises or falls as x increases.

• Relative Maximum/Minimum: Highest/lowest points in a local region.

• Example: For shown in a graph, identify intervals where $f(x)$ is increasing or decreasing.

Transformations of Functions

Transformations shift, stretch, compress, or reflect graphs.

• Vertical Shift: shifts up/down.

• Horizontal Shift: shifts right/left.

• Reflection: reflects over x-axis.

• Example: shifts right by 2 units.

Polynomial and Rational Functions

Polynomial Functions

Polynomials are expressions of the form . Their graphs have smooth, continuous curves.

• Degree: Highest power of x.

• Zeros: Values of x where .

• End Behavior: Determined by leading term.

• Example:

Rational Functions

Rational functions are ratios of polynomials. Key features include vertical and horizontal asymptotes.

• Vertical Asymptote (VA): Values of x where denominator is zero.

• Horizontal Asymptote (HA): Determined by degrees of numerator and denominator.

• Example: has VA at .

Factoring and Zeros

Factoring polynomials helps find zeros and simplify expressions.

• Rational Zero Theorem: Possible rational zeros are factors of constant term over leading coefficient.

• Example: For , possible rational zeros are .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form . They model growth and decay.

• Example:

• Applications: Compound interest, population growth.

Logarithmic Functions

Logarithms are the inverses of exponentials. means .

• Properties:

• Example: Solve ; .

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations can be solved by substitution, elimination, or matrix methods.

• Example: Solve and .

Matrices and Determinants

Matrices organize data and can be used to solve systems. The determinant helps determine if a system has a unique solution.

• Matrix: Rectangular array of numbers.

• Determinant: Scalar value from a square matrix.

• Example: For , .

Gaussian and Gauss-Jordan Elimination

These are systematic methods for solving systems using row operations.

• Row Operations: Swap, scale, or add rows to simplify.

• Example: Reduce to row-echelon form.

Sequences, Induction, and Probability

Sequences

A sequence is an ordered list of numbers, often defined by a formula.

• Arithmetic Sequence:

• Geometric Sequence:

• Example: Find first four terms of .

Series and Summation

The sum of terms in a sequence is called a series.

• Example:

Probability

Probability measures the likelihood of an event.

• Formula:

• Example: Probability of rolling a 3 on a six-sided die:

Conic Sections

Types of Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas, each defined by a specific equation.

• Circle:

• Parabola:

• Ellipse:

• Hyperbola:

Tables

Example: Rational Zero Candidates

The Rational Zero Theorem provides a list of possible rational zeros for a polynomial.

Additional info: Table entries inferred from synthetic division and remainder theorem context.

Additional Info

• Odd and even functions: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• One-to-one functions: Pass the horizontal line test.

• Logarithmic properties: Used to simplify and solve equations.

• Applications: Problems include compound interest, population growth, and mixture problems.