Algebraic Expressions and Exponents

Simplifying Expressions and Eliminating Negative Exponents

Algebraic expressions often involve exponents, fractions, and roots. Simplifying such expressions requires applying the laws of exponents and algebraic manipulation.

• Law of Exponents: For any real numbers a and b, and integers m and n:

• Example: Simplify

  • Apply the exponent to each term inside the parentheses.

  • Combine like terms and eliminate negative exponents.

Combining and Simplifying Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplification involves finding common denominators and reducing the expression.

• Example:

  • Find a common denominator for the terms in the numerator and denominator.

  • Simplify the resulting expression.

Radicals and Rationalization

Rationalizing Denominators and Numerators

Rationalizing involves removing radicals from the denominator or numerator of a fraction. This is often done by multiplying by a conjugate or an appropriate form of 1.

• Rationalizing the Denominator: To rationalize , multiply numerator and denominator by the conjugate .

• Rationalizing the Numerator: For , multiply numerator and denominator by .

• Example:

  • Rationalize by multiplying by the conjugate .

Linear Equations and Functions

Point-Slope and Slope-Intercept Forms

Linear equations can be written in several forms, most commonly point-slope and slope-intercept forms.

• Point-Slope Form: , where is a point on the line and is the slope.

• Slope-Intercept Form: , where is the slope and is the y-intercept.

• Example: The equation of a line through with slope is .

• Finding the Equation from Two Points: Use the slope formula , then substitute into point-slope form.

• Finding Slope and Intercept: Rearrange to to identify slope and y-intercept.

Linear Growth and Decay Models

Modeling with Linear Functions

Linear models describe situations where a quantity increases or decreases at a constant rate.

• General Linear Model: , where is the initial amount, is the rate of change, and is time.

• Example:

  • Increasing:

  • Decreasing:

Quadratic Functions

Standard Form, Vertex, and Intercepts

Quadratic functions have the form . Their graphs are parabolas, and key features include the vertex, y-intercept, and x-intercepts.

• Vertex: The vertex of is at .

• Y-Intercept: The y-intercept is .

• X-Intercepts: Solve using the quadratic formula:

• Graph: The parabola opens upward if and downward if .

• Example: For :

  • Vertex:

  • Y-intercept: $8$

  • X-intercepts: Solve

Factoring Quadratic Functions

Factoring expresses a quadratic as a product of two binomials, which is useful for finding x-intercepts.

• General Form:

• Example:

Table: Quadratic Function Properties

Additional info: Table entries for x-intercepts of and require solving the quadratic formula explicitly.