Algebraic Expressions and Simplification

Simplifying Rational and Radical Expressions

Algebraic expressions often require simplification by applying exponent rules, combining like terms, and rationalizing denominators.

• Exponent Rules: When multiplying or dividing like bases, add or subtract exponents respectively.

• Radical Simplification: Simplify radicals by factoring out perfect squares and rationalizing denominators.

• Example:

  • Simplify

  • Apply exponent rules and simplify each variable separately.

Combining Like Terms and Distributive Property

Expressions can be simplified by combining like terms and using the distributive property to expand or factor expressions.

• Like Terms: Terms with the same variable and exponent can be combined.

• Distributive Property:

• Example: simplifies to

Polynomial Operations

Multiplying and Factoring Polynomials

Polynomials can be multiplied using the distributive property or special products, and factored by grouping or using formulas.

• Multiplying Binomials: Use FOIL (First, Outer, Inner, Last) for .

• Factoring Quadratics: Find two numbers that multiply to and add to in .

• Example: Factor as

Solving Equations

Linear, Quadratic, and Radical Equations

Equations can be solved by isolating the variable, factoring, or applying the quadratic formula. Always check for extraneous solutions, especially with radicals.

• Quadratic Formula:

• Square Root Method: If , then

• Example: Solve yields

Absolute Value Equations

Absolute value equations have two possible solutions, one for the positive and one for the negative case.

• General Form: implies or

• Example: gives or

Functions and Graphs

Identifying Functions and Their Properties

A function assigns exactly one output to each input. The vertical line test can determine if a graph represents a function.

• Vertex of a Parabola: For , the vertex is

• Example: has vertex

Evaluating and Composing Functions

To evaluate a function, substitute the input value into the function's formula. Composition involves plugging one function into another.

• Example: If , then

Transformations of Functions

Transformations include translations, reflections, stretches, and compressions.

• Vertical Shift: shifts up by units.

• Horizontal Shift: shifts right by units.

• Reflection: reflects over the x-axis.

Linear Functions and Applications

Writing Equations of Lines

The slope-intercept form of a line is , where is the slope and is the y-intercept.

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals.

• Example: Line through parallel to is

Linear Models and Applications

Linear functions can model real-world situations, such as depreciation of assets.

• General Linear Model:

• Example: A mower purchased for depreciates by $800V(t) = -800t + 12,000$

Operations with Functions

Function Arithmetic

Functions can be added, subtracted, multiplied, or divided, provided the domain allows.

• Example: If and , then

Inequalities and Absolute Value

Solving Inequalities

Inequalities are solved similarly to equations, but the direction of the inequality reverses when multiplying or dividing by a negative number.

• Example:

• Factor:

• Solution: or

Summary Table: Key Algebraic Concepts

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Graphical questions (e.g., determining if a graph is a function) rely on the vertical line test.

• Transformation descriptions are based on standard function transformation rules.