Equations and Their Solutions

Factoring and Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two. Solving them is a fundamental skill in College Algebra, and several methods are commonly used.

• Factoring: Expressing the quadratic as a product of two binomials and setting each factor to zero to solve for the variable.

• Square Root Property: If , then .

• Completing the Square: Rewriting the equation in the form and solving for .

• Quadratic Formula: For , the solutions are given by:

Example: Solve by factoring: .

Discriminant and Number of Solutions

The discriminant of a quadratic equation is . It determines the nature and number of solutions:

• If : Two distinct real solutions.

• If : One real solution (a repeated root).

• If : Two complex (non-real) solutions.

Example: For , (two complex solutions).

Solving Higher Degree Polynomial Equations

Polynomials of degree greater than two can sometimes be solved by factoring or by using the zero factor property: If , then or .

• Look for common factors or patterns (e.g., difference of squares, sum/difference of cubes).

• Set each factor equal to zero and solve for the variable.

Example: Solve : .

Absolute Value Equations

Equations involving absolute value require considering both the positive and negative cases:

• For , or .

• Isolate the absolute value before splitting into cases.

Example: Solve : or or .

Inequalities and Their Graphs

Solving and Graphing Inequalities

Inequalities can be solved similarly to equations, but the direction of the inequality reverses when multiplying or dividing by a negative number. Solutions are often represented on a number line or in interval notation.

• Graph solutions on a number line, using open or closed circles to indicate inclusion or exclusion of endpoints.

• Express solutions in interval notation, e.g., .

• Special cases: Be careful with fractions and compound inequalities.

Example: Solve : .

Graphing Solutions and Interval Notation

After solving an inequality, represent the solution set visually and in interval notation.

• For , graph an open circle at 3 and shade to the right; interval notation: .

• For , graph a closed circle at -1 and shade to the left; interval notation: .

Relations and Functions

Identifying Functions

A function is a relation in which each input (domain value) corresponds to exactly one output (range value). A relation can be represented as a set of ordered pairs, a table, a graph, or an equation.

• Use the vertical line test on a graph: If any vertical line crosses the graph more than once, it is not a function.

• For a set of ordered pairs, check that no input value is paired with more than one output.

Example: The set {(1,2), (2,3), (3,4)} is a function; {(1,2), (1,3)} is not.

Evaluating Functions

To evaluate a function at a given value, substitute the input into the function's formula.

• Given , find : .

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Evaluate by determining which interval the input belongs to, then using the corresponding formula.

Example: ; , .

Difference Quotient

The difference quotient is a formula that measures the average rate of change of a function over an interval. It is foundational for calculus.

• Formula: , .

• Simplify the expression as much as possible.

Example: For , the difference quotient is .

Properties and Analysis of Functions

Function Properties

Analyzing a function involves determining several key properties:

• Domain: All possible input values (x-values).

• Range: All possible output values (y-values).

• x-intercepts: Points where the graph crosses the x-axis ().

• y-intercept: Point where the graph crosses the y-axis ().

• Intervals of increase/decrease: Where the function is rising or falling as x increases.

• Relative maximum/minimum: Highest or lowest points in a local region of the graph.

• Even/Odd/Neither:

  • Even: for all in the domain (symmetric about the y-axis).

  • Odd: for all in the domain (symmetric about the origin).

  • Neither: Does not satisfy either condition.

Example: is even; is odd; is neither.

Summary Table: Function Properties

The following table summarizes key properties to analyze for any given function:

Applications

Understanding these concepts is essential for graphing functions, solving real-world problems, and preparing for further study in mathematics.

Additional info: Some context and examples have been inferred and expanded for clarity and completeness, as the original document was a syllabus or checklist of learning objectives.