College Algebra Final Exam Review: Comprehensive Study Notes

Equations, Inequalities, and Problem Solving

Linear and Quadratic Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable(s) that satisfy the given equation.

• Linear Equations: Equations of the form $ax + b = c$ can be solved by isolating $x$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $2x + 3 = 7$ $2x = 4 \Rightarrow x = 2$

• Compound Inequalities: Solve and express solutions in interval notation.

Word Problems and Applications

Translating real-world scenarios into algebraic equations is essential for problem solving.

• Mixture Problems: Use systems of equations to solve for unknown quantities.

• Geometry Problems: Apply algebraic methods to find lengths, areas, and other measurements.

• Example: "How much of an alloy that is 20% copper should be mixed with 300 ounces of an alloy that is 50% copper to get an alloy that is 30% copper?"

Graphing

Functions and Their Graphs

Graphing is a visual representation of equations and functions, showing the relationship between variables.

• Linear Functions: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

• Quadratic Functions: $y = ax^2 + bx + c$; the graph is a parabola.

• Vertex Form: $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex.

• Example: Find the slope of the line passing through $(4, 5)$ and $(8, 4)$: $m = \frac{4-5}{8-4} = -\frac{1}{4}$

Graphing Inequalities

Graphing inequalities involves shading regions that satisfy the inequality.

• Linear Inequality: $3x + 2y > 6$

• System of Inequalities: Graph each inequality and find the intersection region.

Solving Systems of Linear Equations

Methods of Solution

Systems of equations can be solved using substitution, elimination, or graphing.

• Substitution: Solve one equation for one variable and substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Graphing: Plot both equations and find the intersection point.

• Example: Solve:

  • $x + y = 4$

  • $2x - y = 1$

  Solution: Add equations to eliminate $y$.

Exponents and Polynomials

Properties of Exponents

Exponents follow specific rules for simplification.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, and multiplication.

• Degree: The highest power of the variable.

• Example: $x^3 - 2x^2 + 4x - 5$ is a cubic polynomial.

Factoring Polynomials

Factoring Techniques

Factoring is rewriting a polynomial as a product of simpler polynomials.

• Common Factor: $ax + ay = a(x + y)$

• Quadratic Factoring: $x^2 + bx + c = (x + p)(x + q)$ where $p$ and $q$ are roots.

• Special Products: $a^2 - b^2 = (a - b)(a + b)$

Rational Expressions

Simplifying and Operations

Rational expressions are fractions involving polynomials.

• Simplify: Factor numerator and denominator, then reduce common factors.

• Operations: Addition, subtraction, multiplication, and division follow fraction rules.

• Domain: Values of the variable that do not make the denominator zero.

Radicals and Rational Exponents

Properties and Simplification

Radicals and rational exponents are alternative ways to express roots and powers.

• Definition: $\sqrt[n]{a} = a^{1/n}$

• Operations: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

• Rationalizing: Remove radicals from the denominator.

Quadratic Equations and Functions

Solving and Graphing Quadratics

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Vertex: The point $(h, k)$ where the parabola changes direction.

• Axis of Symmetry: $x = h$ for $y = a(x-h)^2 + k$

• Discriminant: $b^2 - 4ac$ determines the number of real solutions.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $y = a b^x$.

• Growth and Decay: Used to model population, finance, and other phenomena.

• Example: $y = 2^x - 2$

Logarithmic Functions

Logarithms are the inverse of exponentials.

• Definition: $y = \log_b x$ means $b^y = x$

• Properties: $\log_b (xy) = \log_b x + \log_b y$

Conic Sections

Types and Equations

Conic sections include circles, ellipses, parabolas, and hyperbolas, each with a standard equation.

• Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: $y = ax^2 + bx + c$

• Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

Sequences, Series, and the Binomial Theorem

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers following a specific pattern.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Example: Find the sixth term of a geometric sequence with $a_1 = 5$ and $r = 2$: $a_6 = 5 \cdot 2^{5} = 160$

Tables: Example of a Data Table

U.S. Postage Stamp Data Table

This table shows the cost of U.S. postage stamps over time, useful for modeling with linear or quadratic functions.

Year	Cost (cents)
1971	8
1974	10
1978	15
1981	18
1985	22
1988	25
1991	29
1994	32
1997	32
2002	37

Scientific Notation

Calculating and Converting

Scientific notation expresses numbers as a product of a coefficient and a power of ten.

• Format: $a \times 10^n$

• Example: $7,000,000 \times 0.0001 = 700$

Additional info:

• Some problems require calculator use for graphing, finding intersections, and numerical solutions.

• Questions cover all major College Algebra topics, including equations, inequalities, graphing, systems, polynomials, rational expressions, radicals, quadratics, exponentials, logarithms, conic sections, sequences, and series.