Review of Basic Algebra Skills

Order of Operations and Simplification

Understanding the order of operations is essential for simplifying algebraic expressions. The standard order is Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

• Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

• Combining Like Terms: Add or subtract coefficients of terms with the same variable and exponent.

• Distributive Property:

• FOIL Method: Used for multiplying two binomials: First, Outer, Inner, Last.

Example: Simplify and .

Additional info: The negative sign outside the exponent only applies after the exponent is evaluated.

Combining Like Terms

When simplifying expressions, combine terms with the same variable and exponent.

• Example:

Factoring

Factoring Out a Common Term

Factoring is the process of writing an expression as a product of its factors.

• Example:

Factoring the Difference of Squares

The difference of squares formula is .

• Example:

Factoring Trinomials

To factor trinomials of the form , find two numbers that multiply to and add to .

• Example:

Functions

Definition and Notation

A function is a relation that assigns exactly one output (dependent variable) for each input (independent variable).

• Notation: means "y equals f of x".

• Output:

• Dependent variable: Output ()

• Independent variable: Input ()

Common Error: does NOT mean "f times x"; it means the function evaluated at .

Evaluating Functions

To evaluate a function at a specific value, substitute the value for in the function.

• Example: Given , find , , .

Function Form and Graphing

Solving for the Dependent Variable

To express a relation in function form, solve for the dependent variable (usually ) in terms of the independent variable ().

• Example: → Solve for .

Domain and Range

Definitions

• Domain: The set of possible input values (-values).

• Range: The set of corresponding output values (-values).

Interval Notation

Interval notation is used to represent domains and ranges.

• Closed Interval: includes all real numbers such that .

• Open Interval: includes all real numbers such that .

• Half-Open Interval: or includes all real numbers such that or .

Example: Express "Harry's GPA is more than 2.0, but at most 4.0" in interval notation: .

Visualizing Functions

Types of Output

• Positive Output:

• Negative Output:

• Zero Output:

• Constant Function: Output value is the same for every input.

• Increasing Function: increases as increases.

• Decreasing Function: decreases as increases.

Example: Given a graph, identify intervals where the function is increasing or decreasing, and estimate values.

Average Rate of Change

Definition and Formula

The average rate of change of over an interval measures how much changes, on average, for each one-unit change in .

• If the slope is positive, output increases as input increases.

• If the slope is negative, output decreases as input increases.

• If the slope is zero, output is constant for all input.

Example: Find the average rate of change in LeBron James's scoring from 2010 to 2022.

Linear Functions

Definition and General Equation

A linear function has a constant rate of change and its graph is a straight line.

• General Equation:

• Output = Initial Value + Rate of Change × Input

Slope-Intercept Form:

• m: Slope (rate of change)

• b: y-intercept (starting point)

Slope (m)

• Gives the rate of change of with respect to .

• Tells how much changes for a one-unit change in .

• The sign of determines whether the line rises or falls.

• The larger the magnitude of , the steeper the graph.

Y-Intercept (b)

• Where the graph crosses the y-axis.

• Represents the initial value of the output.

Graphing a Linear Equation

• Plot the y-intercept ().

• Use the slope () to find the next point.

• Connect the points to form a line.

Example: Graph a line with slope $2-3$.

Writing a Linear Equation

• Find the slope.

• Plug in , , and into and solve for .

• Write the equation in form.

Example: Given points and , find the equation of the line passing through them.

Applications of Linear Functions

Interpreting Slope and Y-Intercept

• Slope: Represents the rate of change in context (e.g., cost per year, salary increase per year).

• Y-Intercept: Represents the starting value (e.g., initial cost, starting salary).

Example: If models a loan balance, is the initial loan amount, $250$ is the amount paid off per year.

Domain and Range in Applications

• Domain: Possible input values (e.g., years of service, time).

• Range: Possible output values (e.g., salary, cost).

Example: If an employee starts at salary and receives annual increases, domain is years, range is .

Summary Table: Key Concepts