BackCollege Algebra: Comprehensive Study Notes and Practice Problems
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Review of Algebra
Solving Linear Equations
Linear equations are equations of the first degree, meaning the variable is raised only to the first power. Solving these equations involves isolating the variable on one side.
Key Point: Combine like terms and use inverse operations to solve for the variable.
Example: Solve
Solving for a Variable in Formulas
Sometimes, you need to solve for a specific variable in a formula involving multiple variables.
Key Point: Rearrange the equation to isolate the desired variable.
Example: Solve for
Equations & Inequalities
Word Problems and Applications
Translating real-world scenarios into algebraic equations is a key skill. Identify variables, set up equations, and solve for the unknowns.
Key Point: Assign variables, write equations based on the problem, and solve step by step.
Example: If a taxi ride costs miles
Graphs of Equations
Graphing Linear Equations
Linear equations can be graphed by finding the y-intercept and using the slope to plot additional points.
Key Point: The slope-intercept form is , where is the slope and is the y-intercept.
Example: Graph Slope: , y-intercept: $2$
Finding Slope and Intercepts
Key Point: Slope between two points and is .
Example: Points and :
Functions
Definition and Identification
A function assigns exactly one output to each input. Vertical line test: if any vertical line crosses the graph more than once, it is not a function.
Key Point: Each input (x-value) has only one output (y-value).
Example: The set {(2, 6), (2, 4)} is not a function because 2 maps to both 6 and 4.
Evaluating and Simplifying Functions
Key Point: Substitute the input value into the function and simplify.
Example: If , then
Domain of a Function
Key Point: The domain is the set of all input values for which the function is defined.
Example: For ,
Polynomial Functions
Degree and Turning Points
Key Point: The degree of a polynomial is the highest power of . The maximum number of turning points is one less than the degree.
Example: A polynomial with 4 turning points must have at least degree 5.
Rational Functions
Asymptotes and Intercepts
Key Point: Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at that point). Horizontal asymptotes depend on the degrees of numerator and denominator.
Example: For , vertical asymptotes at and .
Type | How to Find |
|---|---|
Vertical Asymptote | Set denominator = 0 |
Horizontal Asymptote | Compare degrees of numerator and denominator |
Exponential & Logarithmic Functions
Exponential Equations
Key Point: Exponential equations have the form .
Example: because
Logarithmic Equations
Key Point: means .
Example:
Sequences, Series, & Induction
Arithmetic and Geometric Sequences
Key Point: Arithmetic sequence: Geometric sequence:
Example: If , , then
Applications and Word Problems
Compound Interest
Key Point: , where is the amount, is the principal, is the rate, is the number of times compounded per year, and is time in years.
Example: , , ,
Summary Table: Key Concepts
Topic | Key Formula/Fact |
|---|---|
Linear Equation | |
Slope | |
Quadratic Vertex | |
Exponential | |
Logarithm | |
Compound Interest |
Additional info: These notes are based on a comprehensive exam review covering equations, inequalities, graphing, functions, polynomials, rational functions, exponentials, logarithms, and applications, as typically found in a College Algebra course.
